Welcome! for Contents and Teacher Notes. Probability 1. Probability 2. Experimental Probabilities 1. Experimental Probabilities 2.

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1 Welcome! This is the first in a series of teaching aids designed by teachers for teachers at level 7/. The worksheets are designed to support the delivery of the National Curriculum in a variety of teaching and learning styles. They are not designed to take the pedagogy away from the teacher. The worksheets are centred around the shown level, but spiral from the level below to the level above. Consult the National Numeracy Strategy for definitive National Curriculum levels. They can be used by parents with the support of the on-line help facility at Contents and Teacher Notes. Pages 3/. Probability 1. Revision of level 5 and level single event probability, including the probability of events not happening. Pages 5/. Probability. Mutually exclusive and exhaustive events. Pupils have to be familiar with fraction addition and subtraction as well as with decimal addition and subtraction. The addition (OR) rule. Pages 7/. Experimental Probabilities 1. Assigning probabilities. Finding the expectation of a probability. Page 9. Experimental Probabilities. Experiments to compare theory and experiment. Pupils will use the experimental probability diagrams (p 10) to find out experimental probabilities and compare them with theoretical probabilities. Page 10. Experimental Probability Diagram. A grid for recording experimental probability for an event. After 50 trials read off the experimental probability. This could be used for some of the level 5 probability. Photocopy this sheet back to back so pupils will only need one sheet to complete Experimental Probabilities. Pages 11/1. Experimental Probabilities 3. Relative frequency and relative frequency graphs. A pictorial way to show relative frequency "settling down" to theoretical probability with the greater the number of trials. "10 trials" has been chosen so as to fit on 1 piece of A graph paper landscape (long ways). Pages 13/1. Pages 15/1. Pages 17/1. Pages 19/0. Pages 1/. Page 3. Lists and Possibility Spaces. Revision of the Level descriptors. Tree Diagrams (Independent Events). Complete the tree diagram by adding the appropriate probabilities and then solve the questions. Drawing tree diagrams from worded questions. 'AND' and 'OR' Rules. The multiplication rule (AND rule). Questions that involve the AND\ OR rules. Substitution (Negative Numbers and Fractions). Before pupils can plot linear functions they must be comfortable with this type of substitution, with and without a calculator. The formulae chosen are typical of the formulae they will meet around the school. Plotting Linear Functions. Revision of all the level skills. Finding the gradient and y intercept of a line. Rearranging linnear equations. Finding the coordinate of the mid-point of a line segment. Finding the Equation of a Straight Line. Given two coordinates, pupils have to find the equation of the line. Scales are changed on the axes so that pupils realise they have to read off axes and not count squares. Level 7/ Pack 1. Page 1.

2 Page. Finding Equations of Straight Line Graphs 1. The lines are drawn already, pupils have to find the equations. Pages 5/. Finding Equations of Straight Line Graphs. As above. Pages Linear Function Snap Cards. These cards can be used in a variety of ways. They may be used as a simple pairing exercise to stick in the book. They may be used in a game of "pairs", cards are turned upside down, two are turned over. If they match you win them. Use sets of each sheet and play "snap!" The list is as endless. Pages 31/3. Scatter Graphs 1. Plotting scatter graphs and finding the line of best fit by inspection. The line of best fit is then used to find some estimations. Pages 33/3. Scatter Graphs. Scatter graphs are extended by finding the mean of the scores which is plotted to aid positioning the line of best fit. Then the equation of lines of best fit have to be found. This equation is used to find further estimates of values. Pages 35/3. Pages 37/3. Pages 39/0. Pages 1/. Pages 3/. Direct Proportion. Finding the constant of proportionality to solve direct proportion questions. Quadratic Sequences. Revision of finding the n th term of linear sequences. Simple and more difficult quadratic sequences. Finding Quadratic Functions (Difference Method). Only required for coursework. This is a useful sheet to do the week before attempting coursework with a class. It shows pupils how to derive the formula for a quadratic using the difference method. This enables them to score higher marks in strand iii at AT 1. Note: this is not a justification of a formul Practical Quadratic Number Patterns. Looking at practical patterns that derive quadratic functions. These are in the style of "Matchsticks" investigations pupils may meet through coursework. Investigations (Quadratic Patterns). Some investigations that again derive quadratic functions. The first set is grouped together through a football theme, each one is an investigation so all don't have to be completed. The onions investigation is a very good one to stretch the more able and keep the rest going! It is also quite simple to justify the quadratic (see above). Regroup the onions into squares, each diagram will make squares n and (n + 1) in dimension. By working out the bracket and adding them all together it is fairly simple to derive n + n + 1. For the very able, the first pattern can be extended into 3 dimensions leading to the cubic / 3 n 3 + n + / 3 n + 1!!! Copyright in Worksheets. Fisher Educational Ltd Copyright in the worksheets belongs to Fisher Educational Ltd. Each purchase of the worksheets represents a licence to use and reproduce the worksheets as set out in the Terms and conditions shown on the 10ticks website. '10TICKS', and '10TICKS.co.uk' and/or other 10TICKS services referenced on this web site or on the Worksheets are trademarks of Fisher Educational Ltd in the UK and/or other countries. Details of copyright ownership in the clip art used in these worksheets: Copyright in the clip art used entirely in this pack is owned by Nova Development Corporation, California, USA. Level 7/ Pack 1. Page.

3 Single Event (Revision). Probability 1. P (event) = Number of ways the event can occur Total number of outcomes 1). The following spinners are spun. What is the probability of landing on the shaded section for each spinner? a). b). c). d). e). ). Counters are placed in a box. For each of the following boxes, find the probability of a shaded counter being drawn out of the box at random. a). b). c). d). e). 3). If a fair six-sided dice is thrown, what is the probability that the score is :- a). a 5 (five), b). an odd number, c). less than, d). a prime number, e). a square number, f). greater than? ). There are 1 counters in a box numbered 1,, 3,, 5,, 7,, 9, 10, 11, and 1. If one counter is drawn out at random, what is the probability that it is a counter :- a). with an odd number, b). greater than, c). with a square number, d). that is greater than or equal to? 5). I have a pack of playing cards, containing 5 cards. I pick a card at random. What is the probability that the card I select is :- a). a queen, b). a red card, c). a club, d). an ace, e). a black ace, f). the jack of diamonds? ). Writing pads are made in four different colours. In a box there are 3 blue, 10 red, white and 5 green pads. What is the probability when I open the box I randomly pick a :- a). blue pad, b). red pad, c). white pad, d). green pad, e). yellow pad? 7). 1 0 Draw the number line in your book. 1 The number line represents all possible probabilities. Indicate, by arrows labelled "a" to "f" the probabilities for :- a). throwing a HEAD on a coin, b). cutting a spade from a pack of cards, c). October following September, d). you visiting the moon this evening, e). scoring a on a die, f). your teacher winning the lottery. ). Arrange these events in order of which is "most likely" down to which is "least likely":- a). throwing a dice and getting a "", b). being born on a day with "y" in it, c). spinning a coin and it landing "heads", d). somebody having their birthday on Christmas day. Level 7/ Pack 1. Page 3.

4 P (event not happening) = 1 - P (event happening). E.g. In a bag are 7 marbles, are red. One is drawn at random. What is the probability that the marble drawn is not red? P (not red) = 1 - P (red) P (not red) = 1 - / 7 P (not red) = 3 / 7 9). A biased coin has a probability of landing on heads of 0.7. What is the probability of the coin landing on tails? 10). In a bag there are 10 marbles. Four are red. One marble is drawn at random. What is the probability of picking a marble that is not red out of the bag? 11). In a cake shop the probability of a customer buying a cream cake is 5 /. What is the probability of a customer not buying a cream cake? 1). Another biased coin has a probability of landing on tails of 0.3. What is the probability of landing on heads? 13). A fair die is rolled. What is the probability of the die landing on a number that isn't? 1). Year 7 and Year 10 boys play a football match with a penalty shoot-out, so that the teams cannot draw. The probability of Year 7 winning is 0.. What is the probability of Year 10 boys winning? 15). Year and Year 9 girls play a netball match with a penalty shoot-out, so that the teams cannot draw. The probability of Year 9 winning is 0.75, what is the probability of Year winning? 1). Ted plays a game of chess against a computer, it cannot draw. The computer is set to have a 0.3 chance of winning the game. What is the probability that Ted wins the game? 17). Alex plays a game of patience on the computer. The computer is programmed to let the player have a / 9 chance of winning. What is the probability that Alex loses? 1). A bag contains 5 blue discs, white discs, 9 green discs and 10 red discs. One is drawn out at random. What is the probability of not picking :- a). a white disc, b). a green disc, c). a blue disc, d). a red disc? 1 19). 0 1 Draw out the number line in your book. The number line represents all possible probabilities. Indicate, by arrows labelled "a" to "f" the probabilities for :- a). being born on a week day, b). not cutting a club from a pack of cards, c). Tuesday following Monday, d). your teacher laying an egg, e). not scoring a on a dice, f). the next baby born is a girl. 0). Arrange these events in order of which is "most likely" down to which is "least likely":- a). throwing a dice and not getting a, b). being born on a day with an "s" in it, c). not cutting a king from a pack of cards, d). not being born on a week day. Level 7/ Pack 1. Page.

5 Probability. Mutually Exclusive and Exhaustive Events. When events cannot happen at the same time they are called mutually exclusive. E.g. If a coin is thrown P (landing on Heads) = 1 / and P (landing on Tails) = 1 /. It is impossible for the coin to land on Heads and Tails at the same time. These are mutually exclusive. In this example there are no other possible outcomes and these are also called exhaustive events. The probabilities of exhaustive events add up to 1. 1). At a zebra crossing you either wait or walk. The probability at a particular zebra crossing to wait is What is the probability that you will be able to walk? ). John's favourite drawing pin can land point down or point up. The probability for the drawing pin to land point down is 0.. What is the probability it lands point up? 3). The Year football team can win, draw or lose. The probability they win is 0.5, the probability they draw is 0.. What is the probability they lose? ). In the school canteen Jenny can choose from chips, pasta or baked potatoes. The probability she chooses chips is 5 /, the probability she chooses pasta is 1 /. What is the probability she chooses baked potatoes? 5). Billy the cat eats out of a red bowl or a yellow bowl or a green bowl. The probability he eats out of the red bowl is 0.53 and the probability he eats out of the green bowl is What is the probability he eats out of the yellow bowl? ). Year 10 hockey team can win, draw or lose. The probability they win is 3 / 10, the probability they draw is 1 / 5. What is the probability they lose? 7). A shop sells three choices of sandwiches, chicken, egg and salad. The probability a customer chooses chicken is 1 / 3. The probability a customer chooses egg is 1 /. What is the probability that a customer chooses salad? ). A motorist has the choice of 3 car parks in town A, B and C. The probability he parks in Car Park A is 0., the probability he parks in Car Park B is What is the probability he parks in Car Park C? 9). At a junction in the road there are 3 choices which a motorist can take, route 1,, or 3. The probability a motorist takes route 1 is 5 / 9, the probability a motorist takes route is 1 /. What is the probability a motorist takes route 3? 10). A four sided spinner is cut out of card. It is not fair. It is labelled 1,, 3 and. The probability it lands on side 1, is 0., the probability it lands on side is 0.17, the probability it lands on side 3 is 0.. What is the probability it lands on side? Level 7/ Pack 1. Page 5.

6 Addition Rule (OR Rule) When two events are mutually exclusive we can work out the probability of either of them occurring by adding together both the individual probabilities. E.g. A bag contains 5 red marbles, 3 yellow marbles and green marbles. One is picked at random. What is the probability it is a red or yellow marble? P (red marble or yellow marble) = P (red marble) + P (yellow marble ) = 5 / / 10 = / 10 = / 5. 1). Traffic lights can show red, red/amber, amber or green. The probability of showing amber is 0.17, the probability of showing red/amber is 0.1. What is the probability of showing amber or red/amber? ). In a flower contest the probability that a red rose will win is 0.1, the probability that a yellow rose will win is 0.. What is the probability that a red or yellow rose will win? 3). A fair die is rolled. What is probability it landing on a 3 or a? ). There are doors in an office. The probability that Janet enters by door 1 is 0.0 and the probability she enters by door is 0.. What is the probability she enters by either door 1 or? 5). A bag contains blue discs, 1 orange discs and red discs. If a disc is picked at random what is the probability of getting :- a). a blue disc, b). an orange disc, c). a blue or red disc, d). a blue or orange disc, e). not an orange disc, f). a blue, orange or red disc? ). A small box of chocolates contains 3 hard centres, soft centres and 7 chewy centres. What is the probability of picking :- a). a hard centre, b). a hard or soft centre, c). a soft or chewy centre, d). a hard or chewy centre,e). not a soft centre, f). not a soft or hard centre? 7). In a class of thirty pupils 9 play hockey, 1 play football, 5 play rugby and go swimming. If a pupil is selected at random, what is the probability that the pupil will a). play football, b). play hockey or swim, c). play hockey or football, d). not play rugby, e). not swim, f). not play rugby or swim? ). In a cash bag there are six 1 pence coins, eight pence coins, twelve 5 pence coins and four 10 pence coin. If a coin is drawn at random, what is the probability that the coin :- a). is a 5p, b). is a copper coin, c). is not a 10p, d). is a 1p or a 5p, e). is not a copper coin, f). is not a 1p, p or 5p? 9). A box contains 1 blue discs, 10 green discs, 3 yellow discs and 15 red discs. If a disc is picked at random what is the probability of getting :- a). a blue disc, b). a green disc, c). a blue or yellow disc, d). a blue or green disc, e). not a yellow disc, f). a pink disc? 10). A box contains blue discs, green discs, 1 purple discs and 1 red discs. If a disc is picked at random what is the probability of getting :- a). a red disc, b). a green disc, c). a blue or red disc, d). a purple or green disc, e). not a purple disc, f). a blue, green, purple or red disc? Level 7/ Pack 1. Page.

7 Experimental Probabilities 1. Assigning Probabilities. Previously we have seen 3 ways of assigning probabilities. Which of these methods would you use to find :- A -- Use equally likely outcomes. B -- Look back at dat C -- Use a survey or experiment to collect dat 1). The probability a volcano will erupt next year in a particular country. ). The probability a person chosen at random in a school will be right handed. 3). The probability a biased coin will land on Tails when thrown. ). The probability a boy's name will be picked at random out of 30 girls and 30 boys. 5). The probability that a car is stolen on a Friday night in Manchester. ). The probability a drawing pin will land point up when dropped. 7). The probability a fair die will land on when rolled. ). The probability that I win a raffle if I buy 50 out of the 00 tickets on sale. 9). The probability that it will rain on Easter day. 10). The probability that the king of clubs will be chosen from a pack of cards. Using Survey or Experiment. Using our surveys or experiments we can determine a probability for an event. From these probabilities we can predict how many times we would expect a particular event to occur for a certain number of trials. This is the expectation, not what will happen. (The expectation should be a very close model as to what will happen! ). To find the expected number of outcomes multiply the probability of the event by the number of trials. E.g. A police car stops 100 cars at random. 15 drivers did not have road tax. a). What is the probability that a driver doesn't have road tax? 15 = b). If the police car stops another 30 cars, how many might they expect to have no road tax? 15 x 30 = ). A coin is to be thrown in an experiment. How many times would you expect it to land on heads if it is thrown :- a). 00 times, b). 900 times, c). 50 times, d). 9 times? ). A fair die is to be rolled in another experiment. How many times would you expect the die to land on if it is rolled :- a). 10 times, b). 00 times, c). 79 times, d). 1 times? 3). If you carried out the experiments in questions 1). and ). would you expect to get exactly these results? Level 7/ Pack 1. Page 7.

8 ). Here are the results of a survey of cars passing the school. Colour Red Blue Green Yellow Other Number of cars a). What is the probability of the next car passing the school being :- i). red, ii). green, iii). yellow, iv). not blue? b). If 00 cars pass the school, how many would you expect to be red? c). If 30 cars pass the school, how many would you expect to be yellow? d). If 10 cars pass the school, how many would you expect to be blue? 5). A drawing pin is repeatedly dropped in an experiment to see which way up it will land. Here are the results. Outcome Frequency Point up 10 Point down 10 a). What is the probability of the drawing pin landing :- i). point up, ii). point down? b). If the drawing pin is dropped 00 times how many times would you expect it to land point up? c). If the drawing pin is dropped 5 times how many times would you expect it to land point down? ). Jean conducts a survey on pupils favourite snacks. Here are the results:- Snack Crisps Chocolate Fruit Biscuit Other Frequency a). What is the probability of a pupil :- i). liking Crisps, ii). liking Fruit, iii). liking Biscuits, iv). not preferring chocolate? b). If 00 pupils were asked, how many would you expect to prefer biscuits? c). If 150 pupils were asked, how many would you expect to prefer crisps? d). If 70 pupils were asked, how many would you expect to prefer chocolate? 7). In an experiment, 100 seeds are sown, but only 5 germinate. a). What is the experimental probability of a seed :- i). germinating, ii). not germinating? b). If 0 seeds are sown, what number might be expected to germinate? c). If 130 seeds are sown, what number might be expected not to germinate? ). 0 trains arrived at Lostock Station this morning. 1 arrived early, were late and the rest were on time. a). What is the probability that the next train will be:- i). late, ii). early, iii). on time? b). 10 trains arrive in the afternoon, how many might you expect to be early? c). Tomorrow 30 trains are due, how many would you expect to be late? d). This week 7 trains should come through Lostock Station, how many would you expect to be on time? Level 7/ Pack 1. Page.

9 Experimental Probabilities. Theoretical and Experimental Probabilities. You will need an Experimental Probability Diagram and coloured pencils for each question below. 1). You are going to find the experimental probability for throwing a coin and it landing on Heads. On the Experimental Probability Diagram under Event 1 write in "Not a Head (Tails)", under Event write in "Heads". Throw a coin 50 times and record each outcome on the Experimental Probability Diagram. Every time you throw a Head you move to the right, every time you throw a Tail you move to the left. Read off the probability when you reach the bottom and record it. Repeat this experiment again another 3 times. Record the outcomes with a different colour each time. Read off the probability for each set of throws. a). What is the theoretical probability of throwing a coin and it landing on Heads? b). Do all sets of throws agree exactly with the theory? c). Do all sets of throws get exactly the same experimental probability? d). Find the mean of the experimental probabilities. e). Is the mean of the experimental probabilities close to the theoretical probability? ). You are going to find the experimental probability for rolling a die and it landing on a. On the Experimental Probability Diagram under Event 1 write in "Not a ", under Event write in "". Roll the die 50 times and record each outcome on the Experimental Probability Diagram. Every time you roll a you move to the right, every time you roll another number you move to the left. Read off the probability when you reach the bottom and record it. Repeat this experiment again another 3 times. Record the outcomes with a different colour each time. Read off the probability for each set of throws. a). What is the theoretical probability (as a decimal) of rolling a on a die? b). Do all sets of rolls agree exactly with the theory? c). Do all sets of rolls get exactly the same experimental probability? d). Find the mean of the experimental probabilities. e). Is the mean of the experimental probabilities close to the theoretical probability? 3). Find the experimental probability of dropping a drawing pin and it landing point up. On the Experimental Probability Diagram under Event 1 write in "Point Down", under Event write in "Point Up". Drop the drawing pin 50 times and record it on the Experimental Probability Diagram. Every time the drawing pin lands point up you move to the right, every time it lands point down you move to the left. Read off the probability at the bottom and record it. Repeat this experiment again another 3 times. Record the outcomes with a different colour each time. Read off the probability for each set of throws. Find the mean of the experimental probabilities. ). Make up your own experiment. Decide which event you are trying to find an experimental probability for. This will be Event. Use the Experimental Probability Diagram to help find an accurate experimental probability for the event. Level 7/ Pack 1. Page 9.

10 Experimental Probability Diagrams Event 1 Event Event 1 Event Level 7/ Pack 1. Page 10.

11 Experimental Probabilities 3. Comparing Theory to Experiments. If we know the theoretical probability of an event we can compare it to the experimental probability (relative frequency) by setting up a number of trials. By drawing a relative frequency graph, the experimental and theoretical probabilities can be compared over all the trials. The relative frequency is the experimental probability after a given number of trials. or relative frequency of an event = the number of times the event occurs the number of trials Experiment 1. Throw a coin 10 times and record the number of times the coin lands on tails. a). Copy the table below. No. of throws No. of times landed Relative Frequency Relative Frequency (Trials, t) on tails so far (n ). ( P (Tails) = n t ) decimal places b). c). d). e) In the table we need to work out the relative frequency for the first 5 trials, after that every 5 th trial, i.e. the 10 th, 15 th, 0 th, 5 th,... This will make plotting the relative frequency graph much easier. Now throw the coin and fill in the table after each trial. Draw the axes for the relative frequency graph like those below. Draw the theoretical probability line of the coin landing "tails". Plot the relative frequency from the table. Relative Frequency 0.50 Theory line f). 0 No. of trials What do you notice about the relative frequency and the theory line as the number of trials increases? Level 7/ Pack 1. Page 11.

12 Experiment. Roll a die 10 times and record the number of times the die lands on. a). Copy the table below. No. of throws No. of times landed Relative Frequency Relative Frequency (Trials, t) on so far (n ). ( P () = n t ) decimal places b). As before, work out the relative frequency for the first 5 trials, after that every 5 th trial, i.e. the 10 th, 15 th, 0 th, 5 th,... Now roll the die and fill in the table after each trial. c). Draw the axes for the relative frequency graph like those below. d). Draw the theoretical probability line of the die landing on. e). Plot the relative frequency from the table Relative Frequency 0.50 Theory line f). 0 No. of trials What do you notice about the relative frequency and the theory line as the number of trials increases? 1). Make up your own probability experiment. Choose an event that you already know the theoretical probability for. Record the relative frequency in a table and then plot a relative frequency graph for it. What do you notice about the relative frequency and the theory line? In experimental probability terms 00 trials are insignificant. These experiments should have thousands, if not tens of thousands of trials to become significant. The more times you do an experiment the closer the relative frequency will get to the theory. ). Choose an event you don't know the theoretical probability for. Here are some ideas : Which way will a shoe land when pushed off a table? Which way up will a drawing pin land when dropped? Which way will buttered toast land when falling off a plate? Record the relative frequency in a table and then plot a relative frequency graph for it. Assign a probability to your event. Level 7/ Pack 1. Page 1.

13 Lists and Possibility Spaces. 1). Katherine goes to the canteen. She wants a hot and a cold drink with her meal. Write all the permutations she can choose from the list of options below. Hot Drinks: Tea Coffee Hot Chocolate Cold Drinks: Cola Lemonade Milk ). On an activity holiday guests have to choose an indoor and an outdoor sport on the first day. Write all the permutations a guest can choose from the list of options below. Indoor: Darts Pool Snooker Table-tennis Outdoor: Sailing Walking Climbing 3). Sabrina wants to buy a television and video. a). Write all the permutations she can choose from the list of makers below. Television: JVC Sony Toshiba Akai Video: Samsung Sony Panasonic Akai b). If she chooses each at random what is the probability she chooses :- i). both Sony, ii). different makers, iii). the same makers? ). Sanjid and Arthur go to the drinks machine. They buy a drink each. The drinks machine has Coffee, Tea and Hot Chocolate. a). List all the permutations of drinks they could buy. b). It is equally likely as to which drinks they buy, find the probability that they :- i). both get tea, ii). buy one coffee and one tea, iii). buy at least one Hot Chocolate. 5). James has a circular and a triangular spinner numbered as shown. On each spinner it is equally likely to land on any of the numbers. James spins them both. a). List all the possible outcomes he could get. b). Draw a possibility space for the same events. c). He multiplies the two scores together. Find the probability he gets a :- i). score less than 0, ii). score greater than 15, iii). square number ). A fair six sided die and a coin are thrown together. a). List all the different permutations of how the die and coin could land. b). Draw a possibility space for the same events. Using either find the probability of getting :- i). the probability of a Tail and a, ii). the probability of a Head and a, iii). the probability of a Tail and a 7. Level 7/ Pack 1. Page 13.

14 7). A fair spinner has four numbers,, 3, 5. It is spun twice. The sum of the scores is noted. Draw a possibility space and find the probability that :- a). the sum is, b). the sum is or more, c). the sum is 9, d). the sum is a square number. ). a). Two fair coins are thrown. i). List all the different permutations showing how the coins could land. ii). Draw a possibility space for the same events. Using either find the probability of getting :- iii). two Heads, iv). one of each. b). Three fair coins are thrown. i). List all the different permutations showing how the coins could land. ii). Can you draw a possibility space for the three coins? Find the probability of getting :- iii). three Heads, iv). two Tails and one Heads, v). at least one Head. 9). Two fair six sided dice are thrown and the difference between the dice noted. Draw the possibility space. What is the probability that :- a). there is no difference between the two scores, b). there is a difference of 3 between the two scores, c). there is a difference of more than between the scores? 10). In a game a normal fair dice is rolled, then a card is picked at random from 5 cards numbered from 1 to 5. Draw a possibility space. Find the probabilities that :- a). the numbers are both 5, b). the sum of the numbers is, c). the numbers are both the same, d). the sum of the numbers is or 5. 11). Two unbiased dice are numbered 3, 3,,,,. They are thrown together and the total of the two scores is found. Draw a possibility space. Find the probabilities that :- a). the numbers are both the same, b). the sum is 9, c). the sum is an even number, d). the sum is greater than 7. 1). A three sided spinner is numbered 1,, 5. A five sided spinner is numbered, 3,,,. Both are spun at the same time. Draw a possibility space and find the probability that :- a). the sum of the scores is a square number, b). the product of the scores is greater than 0, c). the difference in the score was exactly 1. 13). There are bags of marbles. The first contains 3 red, 1 blue and 3 green, the second contains 1 red, blue and green. A marble from each is removed, draw the sample space. Find the probability of getting :- a). red marbles, b). blue marbles, c). a red and blue marble, d). green marbles, e) a green and blue marble. Level 7/ Pack 1. Page 1.

15 Tree Diagrams (Independent Events). 1). Each morning Bob and Bill catch the same bus. The probability that Bob catches the bus is 0.9 and for Bill it is 0.7. The probabilities are independent of each other. a). Copy and complete the tree diagram. b). Calculate the probability that on a given day :- i). they both catch the bus, ii). Bob catches the bus, but not Bill, iii). neither catch the bus, iv). at least one of them catch the bus. Bob 0.9 catch not catch Bill 0.7 catch not catch catch not catch ). There are 10 books on a shelf in a library. Seven are fiction and three are nonfiction. A member of the public takes a book at random, looks at it, and then replaces it on the shelf. Another member of the public then takes a book at random from the shelf. a). Copy and complete the tree diagram. b). What is the probability the two books taken are :- i). both nonfiction, ii). both fiction, iii). one of each? First book 7 10 fiction nonfiction Second book 3 10 fiction nonfiction fiction nonfiction 3). The Post Office have stated that 9% of all first class letters are delivered the next day. Ron posts letters on Friday. a). Copy and complete the tree diagram. b). What is the probability that :- i). both letters are delivered on Saturday, ii). neither letter is delivered on Saturday, iii). just one of the letters arrives on Saturday, iv). at least one letter is delivered on Saturday? First letter Arrive Saturday 0.9 Not Arrive Saturday Second letter 0.9 Arrive Saturday Not Arrive Saturday Arrive Saturday Not Arrive Saturday ). In a box there are red and green counters, all of the same size. One is drawn and then replaced. A second counter is then drawn. a). Copy and complete the tree diagram. b). What is the probability that :- i). both counters are red, ii). both counters are green, iii). one is red and one green, iv). at least one is green, v). neither is green? First draw 3 5 Red Green Second draw 5 Red Green Red Green 5). A fair coin is thrown three times. Copy and complete the tree diagram, marking the probabilities and outcomes. What is the probability that :- a). all three coins are Heads, b). exactly two coins land on Tails, c). at least one coin lands on Tails, d). no coins land on Heads? First throw Heads Tails Second throw Heads Tails Heads Tails Third throw Heads Tails Heads Tails Heads Tails Heads Tails Level 7/ Pack 1. Page 15.

16 ). On any particular day in Bolton it can be sunny, cloudy or rainy. The probability of it being sunny is 0., cloudy 0.3 and rainy 0.5. Copy and complete the tree diagram for this weekend. Find the probability that :- a). both days are sunny, b). both days are rainy, c). one day is sunny and one day it rains, d). it does not rain at all. Saturday 0. Sunday Sunny 0.5 Cloudy Rainy Sunny Cloudy Rainy Sunny Cloudy Rainy Sunny Cloudy Rainy 7). Four fifths of cars on roads today are of foreign manufacture. A pupil looks out of the window during Science and watches two cars go by. Draw a probability tree for this showing all the probabilities and outcomes. Hence find the probability that :- a). both cars are British, b). both cars are foreign, c). there is one of each, d). at least one car is not British. ). When Alex goes to school, she either goes by car (probability 0.3) or catches the bus (probability 0.7). When she comes home from school she either goes by car (probability 0.), catches the bus (probability 0.5) or walks (probability 0.3). Draw a probability tree for going to and from school, and show all the probabilities and outcomes. Hence find the probability that :- a). she goes and comes back by car, b). both journeys are on the bus, c). at least one of her journeys is by bus, d). she will travel in a vehicle for both, e). at least one of her journeys is by car, f). she walks to and from school. 9). In a bag are 5 red, 3 blue and green counters of equal size. One is picked, the colour noted and then put back into the bag. A second is then drawn. Draw a probability tree for this and show all the probabilities and outcomes. Hence find the probability that :- a). both counters are red, b). one red and one green are drawn, c). at least one blue is picked, d). at least one red is picked, e). neither are green. 10). A couple have three children. It is equally likely that each child is a boy or a girl. Draw a probability tree for this and mark the probabilities and outcomes. Hence find the probability that :- a). all three are boys, b). all three are girls, c). they have girls and a boy, d). they have at least one boy, e). they have no girls. 11). A dice is rolled three times and it is noted whether a six is scored or not. Draw a probability tree for this and mark the probabilities and outcomes. Hence find the probability that :- a). all three are sixes, b). exactly one six is scored, c). at least two sixes are scored, d). at least one six is scored, e). no sixes scored. 1). On a fruit machine there are three drums, each with cherries, lemons and pears. The probability of a cherry is 0.5, the probability of a lemon is 0.1 and the probability of a pear is 0.. Draw a probability tree for this and mark the probabilities and outcomes. Hence find the probability that :- a). all are cherries, b). all are lemons, c). at least one lemon is picked, d). at least one cherry and pear is picked, e). none are pears. Level 7/ Pack 1. Page 1.

17 'AND' and 'OR' Rules. Two events are said to be independent if the outcome of one event does not effect the outcome of the other event. The combined events we have looked at so far are independent events. Multiplication Rule (AND Rule). It is possible to calculate probabilities of combined events without using a tree diagram. This would be the same as multiplying along the branches of a tree diagram. Where there are two independent events A and B, the probability of A and B occurring is the probability of A multiplied by the probability of B. P(A and B) = P(A) x P(B) N.B. The rule is similar if there are 3 or more events. 1). Jane goes shopping. The probability she buys a CD is 0.. The probability she buys a magazine is 0.. The events are independent. What is the probability she buys a CD and a magazine? ). In a football match the probability that Michael scores is 0.. The probability that James scores is 0.. The events are independent. What is the probability that Michael and James score in the football match? 3). A coin and a normal fair die are thrown. Calculate the probability of getting :- a). a Tail and a five, b). a Head and a two. ). Two fair dice are rolled. Calculate the probability of rolling a double six ( and ). 5). For each of the first questions draw a number line from 0-1. Mark on the number line the position of P(A), P(B) and P(A and B) where A and B are the events. a). What do you notice about the position of P(A and B)? b). Will there be any time when P(A and B) isn't the smallest probability? ). A playing card is drawn from the pack of 5 cards. It is replaced, the pack shuffled and another card selected. Find the probability that :- a). the two cards were both red cards, b). the two cards were both diamonds, c). the two cards were both kings, d). the two cards were both black aces, e). a red card then a king were chosen. 7). A bag contains 3 red balls, 5 yellow balls and green balls. A ball is selected at random, the colour noted then it is put back into the bag. A second ball is then selected at random from the bag. Find the probability that :- a). the two balls selected were both red balls, b). the two balls selected were both green balls, c). the first ball was yellow and the second ball was green. d). A third ball is selected. Find the probability all three balls selected are yellow. ). A coin is thrown twice and the results noted. Calculate the probability of :- a). the coin landing on Heads both times, b). the coin landing on a Tail then on a Head, c). the coin landing on a Head and a Tail. d). What is the difference between question b) and c).? The coin is thrown for a third time. e). What is the probability it lands on Heads for all three throws? Level 7/ Pack 1. Page 17.

18 AND and OR Rule. Where the order of the outcome is not specified then we also need to use the OR rule. E.g. Two fair dice are rolled. Calculate the probability of the dice landing on a 5 and a. P(5 and in any order) = P(5 and ) or P( and 5) = ( 1 / x 1 / ) + ( 1 / x 1 / ) = 1 / / 3 = 1 / 1 (Check this on a tree diagram). 1). A die is rolled twice. Calculate the probability of the die :- a). landing on an even number then a 5, b). landing on an even number and a 5, c). landing on a square number and a. ). The letters of the word MISSISSIPPI are written on individual cards and then placed in a bag. Each card is equally likely to be picked. A card is selected at random from the bag, placed back in the bag and a second card picked. Calculate the probability that :- a). the cards picked were both the letter S, b). the cards picked were both the letter P, c). the cards picked were a P then an S, d). the cards picked were an I and an S. 3). A playing card is drawn from the pack of 5 cards. It is replaced, the pack shuffled and another card selected. Find the probability that :- a). the two cards were both clubs, b). the two cards were both red Kings, c). the first card was a two and the second card was red, d). one card was a Jack and the other card was red, e). one card was a club and the other card was a Queen. ). A bag contains red balls, 5 yellow balls and 3 green balls. A ball is selected at random, the colour noted then it is put back into the bag. A second ball is then selected at random from the bag. Find the probability that :- a). the two balls selected were both green balls, b). the first ball was red then the second ball was yellow, c). the two balls selected were a red and a green ball, d). the two balls selected were a yellow and a red ball. 5). The probability Jim is late for work on Monday is 0.3. The probability he is late for work on Tuesday is 0.. Find the probability that :- a). Jim is late on Monday and Tuesday, b). Jim is on time on both Monday and Tuesday, c). Jim is late on only one of the days. ). Emma sits her maths exam. The probability of her passing Paper I is 0. and the probability of her passing Paper II is 0.. Find the probability that :- a). Emma passes both her Paper I and Paper II, b). Emma fails both her Paper I and Paper II, c). Emma passes only one of her Papers. 7). At a golf course the probability of scoring a 'hole in one' at the first hole is 1 / 15. The probability of scoring a 'hole in one' at the second hole is 1 / 10. Find the probability that for these first two holes :- a). a hole in one is scored on both holes. b). a hole in one is scored on the first, but not the second hole, c). a hole in one is scored on only one of the holes. Level 7/ Pack 1. Page 1.

19 Substitution (Negative Numbers and Fractions). 1). Using the formula v = u + at, find the value of v if i). u = 5, a = -, t = 1 ii). u = 1, a = -., t = iii). u = 0, a = -., t = 1 5 iv). u = 1, a = -7.3, t = v). u =, a = -0.3, t = - vi). u = 7, a = -3.1, t = ). Using the formula y = mx + c, find the value of y if i). c = -3, x =, m = 1 ii). c = -3, x = 5, m = iii). c =, x = -7, m = 1 3 iv). c =, x = -., m = 3 v). c = -, x = -3., m = 1 vi). c = -.5, x = -, m = 1 3 vii). c = 3, x = -7, m = - viii). c = -, x = -1., m = - ix). c = -, x = -5.1, m = ). Using the formula a = v - v 1, find the value of a (acceleration) if t i). v 1 = 15, v =, t = 1 ii). v 1 = 1., v = 0., t = iii). v 1 = 10, v = 3.5, t = ). Using the formula s = ut + 1 at, find the value of s if i). u = 5, a = -, t = 1 ii). u = 1, a = -3, t = iii). u = 10, a = -3., t = 1 5 iv). u = 1, a = -, t =. v). u =, a = -0.3, t = vi). u = 7, a = -3.1, t = ). Using the formula v = u - as, find the value of v if i). u = 5, a = -, s = 1 ii). u = 3, a = -3., s = iii). u = 3.5, a = -., s = iv). u = 1 3, a = -7., s = 3 v). u = 3., a = -.5, s = 7 vi). u = 1, a = -3.9, s = 5 3 ). Find the value of b - ac, if i). a = 3, b = 1, c = - ii). a = 3, b = -, c = - iii). a = 1, b = 9, c = iv). a = -1 5, b =, c =. v). a = -5., b =.1, c = -7 vi). a = 1 1, b = -1.9, c = ). Find the value of ab + b, if a i). a =, b = -3 ii). a = 1, b = - iii). a = -1, b = iv). a = -7, b = v). a = 1 3, b = - vi). a = -5., b = -3.1 vii). a = 1 1, b = -1 viii). a = -1, b = 3 Level 7/ Pack 1. Page 19.

20 Substitution (Negative Numbers and Fractions). 1). Using the formula v = u + at, find the value of v if i). u =, a =, t = 1 ii). u = 10, a = -1, t = iii). u = 0, a = -, t = 1 3 iv). u = -1, a = 9, t = v). u = -, a = -5, t = vi). u = 1, a = -0, t = ). Using the formula y = mx + c, find the value of y if i). c = -3, x = 9, m = 1 ii). c = -, x =, m = 3 iii). c = -, x =, m = 3 1 iv). c = 1, x = -, m = 3 v). c = -, x = -3, m = 1 vi). c = -5, x = -, m = 1 3 vii). c = -, x = 15, m = - viii). c =, x = -30, m = - ix). c = -, x =, m = ). Using the formula a = v - v 1, find the value of a (acceleration) if t i). v 1 =, v = 9, t = 1 ii). v 1 =, v = 1, t = 1 iii). v 1 = 10, v = 1, t = 5 3 iv). v 1 = 7, v =, t = 1 v). v 1 =.1, v = 0.7, t = 1 vi). v 1 = 3., v = 0., t = 3 3 ). Using the formula s = ut + 1 at, find the value of s if i). u = 1, a =, t = ii). u = 1, a = 3, t = 10 iii). u = 0, a = -3, t = 1 5 iv). u =, a = -, t = 5 v). u = 3, a = -3, t = vi). u = 1, a = -1, t = ). Using the formula v = u - as, find the value of v if i). u = 1, a = -, s = 1 ii). u = 3, a = -0, s = iii). u =, a = -, s = iv). u = 7, a = -0, s = v). u = 5, a = -0, s = 3 vi). u =10, a = -, s = ). Find the value of b - ac, if i). a = 1, b = -5, c = 9 ii). a = -3, b =, c = 3 iii). a = -5, b = 7, c = 3 iv). a = 1 1, b = 10, c = v). a = -10, b = 9, c = - vi). a = 1 1, b = 11, c = ). Find the value of ab + b, if a i). a = 1, b = - ii). a = 1, b = iii). a = -1, b = 10 iv). a = -1, b = Level 7/ Pack 1. Page 0.

21 A. For each of the following functions find Linear Functions Revision. The general formula for a linear function is y = mx + c where m is the gradient and c is the y intercept. a). the gradient of the line, b). the y intercept. Do not plot any of the functions. 1). y = 5x + ). y = 1 / 5 x + 3). y = x - 5 ). y = 1 / x - 3 5). f(x) -x + ). y = -3x 7). f:x x - 1 ). f(x) - 1 / x + 9). y = -3x + 10). f:x 3 / x - 11). f:x - / 5 x + 7 1). y = -3x +. 13). y = - 3x 1). y = -5 + / 3 x 15). y = - x 1). y = x 17). y =.7 + 3x 1). f(x) -3 - x 19). y = / 3 + x 0). f:x -5 + x 1). f:x / 5 x - 1 / 3 ). f(x) -1 1 / 3 + x 3). y = x - 1 / ). y = - / 7 x 5). y = 3 1 / - 3 / x ). f:x.5x 7). f(x) 5 / 5 x - ). y = / 5 - x B. Rearrange the following linear functions with y the subject of the equation. State i). the gradient of the line, ii). the y intercept. Do not plot any of the functions. 1). 5x y = 0 ). 0 = - y - x 3). x y = 0 ). 0 = 0.5x - y - 5). y - / 5 x - 1 = 0 ). 5 - x + y = 0 7). 3 / x y = 0 ). + y - 7x = 0 9). 0 = y + x - 1 / 3 10). 0 = + y - / 3 x 11). 0 = x + y - 1 1). 0 = 1 - x + y 13). y = x 1). y = -x 15). y = -x 1). x = y ). x = y 1). 3y = x 19). y = x 0). 5y = 1.5x ). y - x = 0 ). 0 = y - x 3)..5x - y = 0 ). y + 0.5x = )..x = y ). 5y = -1.5x 7). y = x ). -1.x = 7y ). x - 3-3y = 0 30). 0 = - y - 3x 31). 0 = - x - 5y 3). x - 5y + = 0 33). x + 3y = 9 3). y - x = 5 35). 3 = y - 10x 3). -9 = y - x 37). 3y x = 0 3). 7 + y - 3x = 0 39). 1-3x + 5y = 0 0). 0 = 1 + x + y C. Plot the following linear functions for values -5 x 5. 1). y = x + ). y = 1 / x - 3 3). y = x - 1 ). y = 1 / 3 x + 5). f(x) x + ). y = -x 7). f:x 3 / x + 1 ). f(x) - / 3 x - 9). x y = 0 10). 0 = 3 - y x 11). y + x - 1 = 0 1). - 3x + y = 0 13). y = -0.x 1). y = x 15). 3y = x 1). 3y = -x ). x + - 3y = 0 1). x + y = -3 19). - x + 5y = 0 0). 0 = x + y Level 7/ Pack 1. Page 1.

22 D. The gradient of a straight line is Change in y = y - y 1 Change in x x - x 1 E.g.1 Find the gradient between (,) and (5, 19). The gradient of a straight line is Change in y = y - y 1 = 19 - = 15 = 5 Change in x x - x E.g. Find the gradient between (-3, ) and (5, ). The gradient of a straight line is Change in y = y - y 1 = - = = 1 Change in x x - x Find the gradient of the line between these sets of coordinates. (Draw a diagram if it helps you). 1). (1,), (5, 10) ). (,3), (,9) 3). (,7), (1,13) ). (,1), (1,) 5). (-,), (,) ). (-,7), (0,15) 7). (-,-), (-1,11) ). (-7,1), (-1,9) 9). (,-), (9, 1) 10). (7,), (13,10) 11). (-3,-), (,7) 1). (3,-3), (11,17) 13). (,-5), (,-1) 1). (-,-13), (0,-1) 15). (-10,-11), (-,7) 1). (-,-), (-,-) E.g.3 Find the gradient between (-1,) and (1,). The gradient of a straight line is Change in y = y - y 1 = - = = -3 Change in x x - x Find the gradient of the line between these sets of coordinates. 17). (5,-1), (3,5) 1). (-5,1), (1,) 19). (-9,-7), (-3,) 0). (-9,), (-3,0) 1). (-3,-1), (,) ). (-7,-), (-,-) 3). (-,), (-,) ). (,5), (-1,-0) 5). (11,-1), (,0) ). (-3,3), (1,5) 7). (-,-1), (-3,1) ). (-1,-19), (-,11) 9). (0,), (-0.5,-1) 30). (,-1), (9,-) 31). (-1,-1), (-1,-10) 3). (-5,3), (-15,-11) E. The mid-point of a line segment joining A (x 1,y 1 ) to B (x,y ) is (x m,y m ) = x 1 + x, y 1 + y Find the coordinate of the mid-point joining the following points 1). (,3) and (, 11) ). (,1) and (, ) 3). (3, 0) and (7, 1) ). (, 7) and (10,13) 5). (0,1) and (, ) ). (1, 0) and (9,1) 7). (, 17) and (1, 5) ). (7, ) and (1, 1) 9). (5, 3) and (19, 19) 10). (1, ) and (1, 10) 11). (1, 5) and (0, 1) 1). (7, 17) and (11, ) 13). (0, 17) and (15, 5) 1). (7, 1) and (9, 1) 15). (, 3) and (, ) 1). (3, 5) and (1, ) 17). (, 13) and (1, ) 1). (7, ) and (, 1) 19). (-5, ) and (1, 1) 0). (7, -) and (, ) 1). (5, ) and (-5, 15) ). (, 11) and (15, -9) 3). (-9, ) and (5, -1) ). (15, -) and (-3, 5) 5). (7, -7) and (-1, 3) ). (-1, 5) and (3, -) 7). (-7, -15) and (13, ) ). (-, ) and (-1, -11) 9). (, -15) and (, -1) 30). (-37, 15) and (-9, -3) Level 7/ Pack 1. Page.

23 Finding the Equation of a Straight Line. Draw appropriate axes for each question, using the scales given. Plot the two coordinates given. Join them with a straight line. Find the equation of the line. Ques. 1-. On the x-axis 1 cm = 1 unit, on the y-axis 1 cm = 1 unit. 1). a). (1,) and (-3,-) b). (,10) and (-,-10) c). (7,10) and (-3,-10). ). a). (,7) and (-1,-) b). (,9) and (-,-9) c). (,) and (-1,-3). 3). a). (3,-3) and (,0) b). (5,7) and (,-) c). (7,) and (-5,). ). a). (,) and (-,0) b). (,) and (-,-) c). (,-) and (-,-10). Ques.5. On the x-axis 1 cm = 1 unit, on the y-axis cm = 1 unit. 5). a). (,-1) and (-,-) b). (,) and (-,0) c). (,) and (-,0). Ques.. On the x-axis cm = 1 unit, on the y-axis 1 cm = 1 unit. ). a). (1,) and (-3,-) b). (,) and (-,-) c). (3,) and (0,-). Ques On the x-axis 1 cm = 1 unit, on the y-axis 1 cm = 1 unit. 7). a). (-3,-9) and (,9) b). (,3) and (,-) c). (,10) and (-3,-11). ). a). (,) and (-3,1) b). (3,11) and (-1,-9) c). (7,) and (-5,0). 9). a). (,-7) and (-,9) b). (0,-3) and (,-11) c). (-,10) and (3,-). Ques On the x-axis 1 cm = 1 unit, on the y-axis cm = 1 unit. 10). a). (,-1) and (,3) b). (,) and (-3,5) c). (0,-5) and (,-3). 11). a). (,) and (,5) b). (,-5) and (-,-) c). (-1,5) and (7,1). Ques. 1. On the x-axis cm = 1 unit, on the y-axis 1 cm = 1 unit. 1). a). (1,-3) and (-3,9) b). (-3,) and (,-11) c). (3,-7) and (-3,11). Ques On the x-axis 1 cm = 1 unit, on the y-axis 1 cm = 1 unit. 13). a). (-,7) and (,3) b). (,0) and (,-) c). (,) and (-,5). 1). a). (1,1) and (-,10) b). (,1) and (3,-) c). (7,) and (-1,10). 15). a). (,) and (,3) b). (-,9) and (1,-9) c). (-,5) and (,). 1). a). (5,7) and (-1,-11) b). (7,-) and (-,10) c). (3,9) and (-,-11). 17). a). (,9) and (3,) b). (,-1) and (-,-7) c). (7,10) and (-5,). 1). a). (5,-9) and (-,1) b). (3,11) and (0,-7) c). (-,) and (,). Ques On the x-axis cm = 1 unit, on the y-axis 1 cm = 1 unit. 19). a). (-,) and (1,-7) b). (-1,11) and (3,9) c). (3,9) and (-3,). 0). a). (,-10) and (-,10) b). (3,-) and (-3,-3) c). (3,7) and (-1,-13). Level 7/ Pack 1. Page 3.