Tutorial 3 - Discrete Probability Distributions

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1 Tutorial 3 - Discrete Probability Distributions 1. If X ~ Bin(6, ), find (a) P(X = 4) (b) P(X 2) 2. If X ~ Bin(8, 0.4), find (a) P(X = 2) (b) P(X = 0) (c)p(x > 6) 3. The probability that a pen drawn at random from a box of pens is defective is 0.1. If a sample of 6 pens is taken, find the probability that it will contain (a) No defective pens (b) 5 or 6 defective pens (c) Less than 3 defective pens 4. An unbiased die is thrown 7 times. Find the probability of throwing at least 5 sixes. 5. A fair coin is tossed 6 times. Find the probability of throwing not more than 4 heads. 6. Assuming that a couple is equally likely to produce a girl or a boy, find the probability that in a family of 5 children there will be more boys than girls. 7. The probability that a shopper chooses Soapysuds when buying washing powder is Find the probability that in a sample of 8 shoppers, the number who choose Soapysuds is (a) exactly 3 (b) more than If the probability that it will rain on any given day in September is 0.3, calculate the probability that in a given week in September, it will rain on (a) Exactly two days (b) At least two days (c) More than half the days (d) Exactly three days that are consecutive 9. In a multiple choice test there are 10 questions and for each question there is a choice of 4 answers, only one of which is correct. If a student guesses at each of the answers, find the probability that he gets (a) None correct (b) More than 7 correct. If he needs to obtain over half marks to pass and the questions carry equal weight, find the probability that he passes % of a box of light bulbs is faulty. What is the largest sample size which can be taken if it is required that the probability that there are no faulty light bulbs in the sample is greater than o.5?

2 11. The probability that a target is hit is 0.3. Find the least number of shots which should be fired if the probability that the target is hit at least once is greater than A die is biased and the probability, p, of throwing a six is known to be less than. An experiment consists of recording the number of sixes in 25 throws of the die. In a large number of experiments the standard deviation if the number of sixes is 1.5. Calculate the value of p and hence determine the probability that exactly three sixes are recorded during a particular experiment. 13. For each of the experiments described below, state, giving a reason, whether a binomial distribution is appropriate. Experiment 1: A bag contains black, white and red marbles which are selected at random, one at a time with replacement. The colour of each marble is noted. Experiment 2: This experiment is a repeat of Experiment 1 except that the bag contains black and white marbles only. Experiment 3: This experiment is a repeat of Experiment 2 except that marbles are not replaced after selection. 14. In two binomial distributions the ratio of the number of independent trials is 5:6, the ratio of the arithmetic means is 2:9 and the ratio of the variances is 32:45. For each distribution, find the probability of success. 15. In the mass production of a certain component it is found that 5% are defective. Components are selected at random and packed in boxes of 10. If one box is selected, calculate the probability that there will be (a) Exactly 2 defectives. (b) More than 1 defective If 2 boxes are selected, calculate the probability that there will be exactly 1 defective among the 20 components. If 150 boxes are selected, estimate the number of boxes which contain no defectives. 16. In a large city 1 person in 5 is left-handed. (a) Find the probability that in a random sample of 10 people (i) Exactly 3 will be left-handed. (ii) More than half will be left-handed. (b) Find the most likely number of left-handed people in a random sample of 12 people (c) Find the mean and the standard deviation of the number of left-handed people in a random sample of 25 people.

3 (d) How large must a random sample be if the probability that it contains at least one lefthanded person is to be greater than 0.95? 17. A crossword puzzle is published in The Times each day of the week, except on Sunday. A woman is able to complete, on average, 8 out of 10 of the crossword puzzles. (a) Find the expected value and the standard deviation of the number of completed crossword puzzles in a given week. (b) Show that the probability that she will complete at least five in a given week is (c) Given that she completes the puzzle on Monday, find the probability that she will complete at least 4 in the rest of the week. (d) Find the probability that in a period of four weeks, she completes 4 or less in only one of the four weeks. 18. An insurance company receives on average 2 claims per week from a certain factory. Assuming that the number of claims follows a Poisson distribution, find the probability that (a) It receives more than 3 claims in a given week (b) It receives more than 2 claims in a given fortnight 19. On average one in 200 cars breaks down on a certain stretch of road per day. Find the probability that on a certain day (a) None of a sample of 250 cars breaks down. (b) More than 2 of a sample of 300 cars break down. 20. The probability that a particular make of light bulb is faulty is The light bulbs are packed in boxes of 100. (a) Find the probability that in a certain box there are (i) No faulty light bulbs (ii) 2 faulty light bulbs (iii) More than 3 faulty light bulbs. (b) A buyer accepts a consignment of 50 boxes if, when he chooses two boxes at random, he finds that they contain no more than two faulty light bulbs altogether. Find the probability that he accepts the consignment. 21. Eggs are packed in boxes of 500. On average, 0.8% of the eggs are found to be broken when the eggs are unpacked. (a) Find the probability that in a box of 500 eggs (i) Exactly 3 will be broken (ii) Less than 2 will be broken.

4 (b) A hypermarket unpacks 100 boxes of eggs. What is the probability that there will be exactly 4 boxes containing no broken eggs? 22. An aircraft has 116 seats. The airline has found that on average 2.5% of people with tickets for a particular flight do not arrive for that flight. If the airline sells 120 seats for a particular flight determine the probability that more than 116 people arrive for that flight. Determine also the probability that there are empty seats on the flight. 23. Fanfold paper for computer printers is made by putting perforations every 30 cm in a continuous roll of paper. A box of fanfold paper contains 2000 sheets. Stage the length of the continuous roll from which the box of paper is produced. The manufacturers claim that faults occur at random and at an average rate of 1 per 240 m of paper. Find the probability that a box of paper has no faults and also the probability that is has more than four faults. Two copies of a report which runs to 100 sheets per copy are printed on this sort of paper. Find the probability that there are no faults in either copy of the report and also the probability that just one copy is faulty. 24. A process for making plate glass produces small bubbles scattered at random in the glass, at an average rate of four small bubbles per 10 m 2. Assuming a Poisson model, find the probability that a piece of glass 2.2 m x 3.0 m will contain (a) Exactly two small bubbles, (b) At least one small bubble (c) At most two small bubbles Show that the probability that five pieces of glass, each 2.5 m x 2.0 m, will be free of small bubbles is e A shop sells a particular make of radio at a rate of 4 per week on average. The number sold in a week has a Poisson distribution. (a) Find the probability that the shop sells at least 2 in a week (b) Find the smallest number that can be in stock at the beginning of a week in order to have at least 99% chance of being able to meet all demands during that week. 26. Lemons are packed in boxes, each box containing 200. It is found that on average, 0.45% of the lemons are bad when the boxes are opened. Find the probabilities of 0, 1. 2 and more than 2 bad lemons in a box. A buyer who is considering buying a consignment of several hundred boxes checks the quality of the consignment by having a box opened. If the box opened contains no bad lemons he buys

5 the consignment. If it contains more than 2 bad lemons he refuses to buy, and if it contains 1 or 2 bad lemons he has another box opened and buys the consignment if the second box contains fewer than 2 bad lemons. What is the probability that he buy s the consignment? Another buyer checks the consignments on a different basis. He has one box opened; if that box contains more than 1 bad lemon he asks for another to be opened and does not buy if the second also contains more than 1 bad lemon. What is the probability that he refuses to buy the consignment? 27. A shopkeeper hires vacuum cleaners to the general public at R5.00 per day. The mean daily demand is 2.6. (a) Calculate the expected daily income from this activity assuming an unlimited number o vacuum cleaners is available. The demand follows a Poisson distribution. (b) Find the probability that the demand on a particular day is (i) 0 (ii) exactly one (iii) exactly two (iv) three or more. 28. Weak spots occur at random in the manufacture of a certain cable at an average rate of 1 per 100 m. If X represents the number of weak spots in 100 m of cable, write down the distribution of X. Lengths of this cable are wound on to drums. Each drum carries 50 m of cable. Find the probability that a drum will have 3 or more weak spots. A contractor buys 5 such drums. Find the probability that two have just one weak spot each and the other three have none. A special drum of this cable, of length 2 km is manufactured. Find the probability that this drum has fewer than 15 weak spots.

S2 QUESTIONS TAKEN FROM JANUARY 2006, JANUARY 2007, JANUARY 2008, JANUARY 2009

S2 QUESTIONS TAKEN FROM JANUARY 2006, JANUARY 2007, JANUARY 2008, JANUARY 2009 SECTION 1 The binomial and Poisson distributions. Students will be expected to use these distributions to model a real-world