Applied Mathematics 12 Selected Solutions Chapter 1

Transcription

1 TLE OF ONTENTS H ELP pplied Mathematics Selected Solutions hapter Tutorial : Experimental and Theoretical Probability Exercises, page 0 hecking Y our Skills The simulation of the 00 trials of each experiment could be done physically using a standard deck of playing cards card would be drawn from the deck, the resulting card is recorded, and the card is then placed back into the deck This process is repeated 00 times The number of times the desired event occurs is counted This number is divided by 00 to obtain the experimental probability The 00 trials could also be simulated using the randin( command on the graphing calculator a ) To use the randin( command, press µ 7 00,, ) e The resulting number is then divided by 00 See the screen below In this trial, the probability is 009 b) Using the randin( command, press µ 7 00, 0, ) e The resulting number is then divided by 00 See the screen below In this trial, the probability is about 0 c) Using the randin( command, press µ 7 00, 0, ) e The resulting number is then divided by 00 See the screen below In this trial, the probability is about 0 d) Using the randin( command, press µ 7 00, 6, ) e The resulting number is then divided by 00 See the screen below In this trial, the probability is 00 opyright 00 Pearson Education anada Inc, Toronto, Ontario SELETED SOLUTIONS

2 TLE OF ONTENTS pplied Mathematics Selected Solutions hapter 9 a) nswers may vary One method could be to toss the two coins 0 times and record the number of times heads occurs This number is divided by 0 to get the experimental probability nother method could be to use the randin( command Press µ 7 0, 0, ) e The resulting number is then divided by 0 See the screens below In this trial, the probability is 0 b) The theoretical probability of getting heads is or 0 c) The results are different The experimental probability from the first screen above is the same as the theoretical probability However, the experimental probability from the second screen above is greater than the theoretical probability a) nswers may vary The lighter side is more likely to appear because gravity will force the heavier side to land face down b) nswers may vary The assumption that there are only equally likely outcomes is reasonable because the faces have the largest area of the coin lthough it is possible for the coin to land on its side, it is unlikely because the side of the coin is very thin compared to the faces The weight of the faces causes the coin to land on one of its faces Tutorial : Generating and Using Sample Spaces Exercises, page 7 0 a) There will be possible outcomes in this sample space For each question there are choices Since there are questions, then the number of outcomes is or b) The probability that the student will score 00% on this quiz is or 00 Since there is only choice for each of the questions, then the total number of outcomes is or This means that there is only out of possible outcomes that will produce a perfect score on the quiz c) The probability that a student will get at least question correct means that the student will get question correct, or questions correct, or questions correct, or questions correct, or questions correct opyright 00 Pearson Education anada Inc, Toronto, Ontario SELETED SOLUTIONS

3 TLE OF ONTENTS pplied Mathematics Selected Solutions hapter P( question correct) = = 6 = P( question correct) = = = P( question correct) = = = P( question correct) = = = 6 P( question correct) = = P(at least question correct) = P( question correct) + P( questions correct) + P( questions correct) + P( questions correct) + P( questions correct) = = = Tutorial : The F undamental ounting Principle Exercises, page a) There are 000 possible outcomes in this sample space for the Pick lottery For each digit there are 0 choices Since there are digits, the number of outcomes is or 000 For Straight Play, the digits must be in the same order as those picked Therefore, there is choice for the first digit, choice for the second digit, and choice for the third digit Thus, the probability of picking digits in the same order is or This means that the odds of winning for Straight Play 000 are in 000 For ox Play, the digits can be in any order as those picked Therefore, there are choices for the first digit, choices for the second digit, and choice for the third digit Thus, the probability of picking digits in any order is or 6 = This means that the odds of winning for ox Play are in 67 a) i) (, ), (, ), (, ) ii) (, ), (, ), (, D), (, ), (, D), (, D) iii) (, ), (, ), (, D), (, E), (, ), (, D), (, E), (, D), (, E), (D, E) b) i) = 6 ii) = iii) = 0 These are double those in part a because the order in which the Ds are picked does not matter For instance, the choices (, ) and (, ) are counted as being different choices, whereas in part a these choices are viewed as the same choice opyright 00 Pearson Education anada Inc, Toronto, Ontario SELETED SOLUTIONS

4 TLE OF ONTENTS pplied Mathematics Selected Solutions hapter c) person could choose Ds from the current top 0 in 0 9 or ways Tutorial : Independent and Dependent Events Exercises, page 0 a) P() = P(drawing marble X in either of the two selections) = b) P( ) = P(drawing marble Y if marble X has already been selected) = c) P( and ) = P() P() = P() P( ) = = d) P( and ) = P() P( ) a) Let represent the event of drawing a diamond Let represent the event of drawing a diamond given that event has occurred P(both diamonds) = P( and ) = P() P( ) = = 7 = 009 b) Let represent the event of drawing a red card Let represent the event of drawing a red card given that event has occurred P(both red) = P( and ) = P() P( ) = 6 = 0 = 0 c) Let represent the event of drawing a face card Let represent the event of drawing a face card given that event has occurred P(both face card) = P( and ) = P() P( ) = = = 009 opyright 00 Pearson Education anada Inc, Toronto, Ontario SELETED SOLUTIONS

5 TLE OF ONTENTS pplied Mathematics Selected Solutions hapter d) Let represent the event of drawing a Let represent the event of drawing a given that event has occurred P(both a ) = P( and ) = P() P( ) = = = 000 e ) Let represent the event of not drawing a club Let represent the event of not drawing a club given that event has occurred P(both not a club) = P( and ) = P() P( ) = 9 = 9 = 09 Tutorial : Mutually Exclusive Events Exercises, page 7 a) and include regions,,,, 6, and 7 b) and include regions,,, 6, 7, and opyright 00 Pearson Education anada Inc, Toronto, Ontario SELETED SOLUTIONS

6 TLE OF ONTENTS pplied Mathematics Selected Solutions hapter c) and include regions,,, 6, 7, and d) or include regions and e) or include regions and 7 f) or include regions and 6 opyright 00 Pearson Education anada Inc, Toronto, Ontario SELETED SOLUTIONS 6

7 TLE OF ONTENTS pplied Mathematics Selected Solutions hapter g) and and include regions,,,, 6, 7, and h) or or include region onsolidating Y our Skills What Should I e ble To Do, page a) FIRST SPIN WIN WIN LOSE SEOND SPIN WIN WIN WIN, WIN WIN, WIN WIN, WIN WIN, WIN LOSE, WIN LOSE, WIN LOSE WIN, LOSE WIN, LOSE LOSE, LOSE b) i) The probability that the spinner lands on win on the first and second spins is or about 0 9 ii) The probability that the spinner lands on win on the first or second spin is or about 09 9 c) i) P() = or about 067 ii) P() = or about 067 iii) P( and ) = 9 or about 0 iv) P( or ) = or about 09 9 d) Yes The results for part c agree with the probabilities from part b e) nswers may vary One solution may be as follows The results from this experiment are: (, 0), (0, 0), (, 0), (0, 0), (0, 0), (, ), (, ), (0, ), (, ), (, 0), (, ), (, ), (, ), (, 0), (0, ) i) The probability that the spinner lands on win on the first and second spin is 6 out of or 0 opyright 00 Pearson Education anada Inc, Toronto, Ontario SELETED SOLUTIONS 7

8 TLE OF ONTENTS pplied Mathematics Selected Solutions hapter ii) The probability that the spinner lands win on the first or second spin is out of or 0 f) lthough the two probabilities are very close in this case, the experimental probabilities are slightly lower than the theoretical probabilities g) s the number of trials increase the experimental probability gets closer and closer to the theoretical probability opyright 00 Pearson Education anada Inc, Toronto, Ontario SELETED SOLUTIONS