12 Math Chapter Review April 16 th, Multiple Choice Identify the choice that best completes the statement or answers the question.

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1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which situation does not describe a discrete random variable? A The number of cell phones per household. B The number of kilometres travelled by car each month. C The number of students in a classroom. D The number of credit cards used by a person 2. What is the expectation for this probability distribution? x P(x) A 0.25 C B 1.0 D Which situation does not describe a uniform distribution? A Cutting a card from a standard deck. B Rolling a 12-sided die. C Counting successes when rolling 6 dice. D A computer randomly generating a number between 1 and A number is randomly chosen from 1 to 9. What is the expected outcome? A 5 C 45 B 4.5 D Which is not a characteristic of a uniform distribution? A B C D All trials are dependent and involve success and failure. All outcomes are equally likely in a single trial. 6. A spinner has 8 sectors, each of which is sized based on their labelled number, from 1 to 8. Which statement is correct? A This is a uniform distribution because it is a single trial. B This is a uniform distribution because each outcome is equally likely. C This is a non-uniform distribution because there is more than one trial. D This is a non-uniform distribution because the sectors have different sizes and therefore different probabilities.

2 7. Six people are randomly selected from seven men and eight women. Which expression represents the probability of four men being selected? A C B D 8. A five-card hand is dealt from a standard deck. Which expression represents the calculation for the expected number of face cards? A C B D 9. Which is a false statement when comparing discrete probability distributions? A Uniform: All outcomes are equally likely. Binomial: Trials are independent. Hypergeometric: Trials are dependent. B Uniform: Binomial: C D Hypergeometric: Uniform: Random variable is the number of items selected. Binomial: Random variable is the number of successful outcomes. Hypergeometric: Random variable is the number of successful outcomes. Uniform: n is the number of items available. Binomial: n is the number of trials. Hypergeometric: n is the size of the population. 10. For a statistics research project, your team recorded the time between successive buses over a one-week period. Which type of probability distribution would be used to calculate the probability that there will be 10 min between buses? A Uniform, since you are considering a single trial. B Binomial, since trials are independent. C Hypergeometric, since trials are dependent. D None of the above, since the random variable is continuous.

3 Matching Match the correct type of probability distribution to each of the following characteristics. A probability distribution may be used more than once or not at all. A uniform distribution B binomial distribution C hypergeometric distribution D uniform or binomial distribution E binomial or hypergeometric distribution 1. all outcomes are equally likely 2. trials are dependent 3. successful outcomes are counted 4. E(X) = np Match the correct variable to each of the following descriptions of parameters of a binomial distribution. A variable may be used more than once or not at all. A n D r B x E q C p 7. the number of trials 8. the number of successful outcomes 9. the probability of success on a single trial 10. the probability of failure on a single trial

4 c) 12 Math Chapter Review April 16 th, 2017 Short Answer 1. To determine the number of printers needed in an office, a study records the number of printing jobs sent to its printers by each employee during a one-hour time period. a) Identify the random variable. b) Calculate the probability of each outcome. Number of Jobs Frequency Probability State whether each scenario can be modelled using a uniform probability distribution. For those that are not uniform, explain why. For each uniform distribution, identify the random variable. a) A number is chosen randomly from 1 to 100. b) Cutting a card from a shuffled deck. c) The colour of automobiles on a highway. d) The number of times a 5 shows when rolling a die 10 times. e) The teacher randomly selects a student s name from a class list. 3. Model each of the following situations using a binomial distribution. Identify the discrete random variable, X; the number of trials, n; the probability of success, p; and the probability of failure, q, in any trial. a) A game consists of rolling a die eight times. You win if the result is 5 or 6. You record the number of wins. b) A stand of 500 trees is infected by a particular insect. The chance of survival is 30%. The number of surviving trees is recorded.

5 4. A bin of 100 travel clocks are tested to see if they work properly, knowing that 5% of them are defective. The clocks are checked until a defective clock is found. a) Explain why this situation cannot be modelled using a binomial distribution. b) Describe how to change this situation so that it can be modelled using a binomial distribution. 5. A poll indicates that 35% of the population supports the current party in power. Twenty Canadians are interviewed. What is the likely number who support the party in power? Justify the formula you used. 6. State whether each of the following situations can be modelled using a hypergeometric probability distribution. Justify your response. a) A box contains 5 red and 7 green balls. One ball is chosen at random and the colour is recorded. b) The faces of a 12-sided die are numbered from 1 to 12. The number of 12s is recorded on 10 rolls of the die. c) A pet store has 6 puppies and 9 kittens. Four are randomly selected to place in the storefront window. The number of kittens is recorded. 7. Each expression represents the probability of a hypergeometric distribution. State the values of the unknowns. a) b) c) d)

6 8. a) Identify one characteristic of a binomial distribution that is shared by the hypergeometric distribution. b) Identify one characteristic that is different. 9. A grocer s bin contains 15 ripe tomatoes and 18 unripened tomatoes. Five tomatoes are selected at random. Show the probability distribution for the number of ripe tomatoes.

Problem 1. The table shows the distribution of household sizes in Canada, based on the 2011 census. Household Size 1 2 3 4 5 6+ Percent of Households 27.6% 34.1% 15.6% 14.3% 5.4% 2.

7 Problem 1. The table shows the distribution of household sizes in Canada, based on the 2011 census. Household Size Percent of Households 27.6% 34.1% 15.6% 14.3% 5.4% 2.9% a) Represent this distribution using a probability histogram. b) What is the expected size of a randomly selected household? 2. A quiz contains five multiple choice questions, each with four possible answers. a) Show the probability distribution for the number of correct answers for someone who guesses at each question.

8 d) 12 Math Chapter Review April 16 th, 2017 b) What is the probability of this person passing? c) Verify the expectation formula for this distribution using the probabilities in the chart. Explain any differences. 3. There are 10 men, 14 women, and 16 children on a harbour cruise boat. Seven passengers are selected at random to win special prizes. a) What is the probability that the winners are all children? b) What is the probability that exactly two are children? c) What is the probability that exactly three are children and two are women? d) What is the expected number of winners who are men, women, and children?

9 Answer Section MULTIPLE CHOICE 1. ANS: B 2. ANS: D 3. ANS: C 4. ANS: A 5. ANS: C 6. ANS: D 7. ANS: A 8. ANS: C 9. ANS: C 10. ANS: D MATCHING 1. ANS: A 2. ANS: C 3. ANS: E 4. ANS: B 5. ANS: A 6. ANS: C 7. ANS: A 8. ANS: B 9. ANS: C 10. ANS: E SHORT ANSWER 1. a) The random variable is the number of printing jobs by employee. Number of Frequency Probability Jobs

10 2. a) Yes. The random variable is a number between 1 and 100. b) Yes. The random variable is any one of the 52 cards in a standard deck. c) No. The colours are not equally likely. d) No. There are 10 trials. e) Yes. The random variable is a student s name from the list. 3. ANS: a) X = the number of 5s or 6s; n = 8; ; b) X = the number of surviving trees; n = 500; p = 0.3; q = ANS: a) Although each trial is independent, this process does not have a set number of trials. b) Test a specific number of clocks instead of stopping when a defective clock is found. 5. ANS: Trials are independent and successes are noted, so this is a binomial distribution. Use the expectation formula, E(X) = np. E(X) = = 7 It is likely that 7 of the 20 who were interviewed support the current party in power. 6. ANS: a) This is not a hypergeometric distribution because it involves only one trial. b) This is not a hypergeometric distribution because the trials are independent. c) This is a hypergeometric distribution because the trials are dependent and successes (kittens) are counted in a finite number of trials. 7. ANS: Compare each expression with. a) a = 9, b = 3 b) c = 7, d = 4 c) e = 6, f = 2 d) g = 9, h = ANS:

For this hypergeometric distribution, n = 33, r = 5, a = 15, and x = 0, 1, 2, 3, 4, 5. Number of Ripe Tomatoes, x Probability, P(x) 0 1 2 3 4 5 9. ANS: Answers may vary.

11 For this hypergeometric distribution, n = 33, r = 5, a = 15, and x = 0, 1, 2, 3, 4, 5. Number of Ripe Tomatoes, x Probability, P(x) ANS: Answers may vary. Examples: a) The number of successes are counted. Alternative: There is a set number of trials. b) Trials are independent with the binomial distribution and dependent with the hypergeometric distribution. Alternative: In the binomial distribution, n represents the number of trials. In the hypergeometric distribution, n represents the size of the population and r represents the number of trials. PROBLEM 1. ANS: a)

12 b) E(X) = 1(0.276) + 2(0.341) + 3(0.156) + 4(0.143) + 5(0.054) + 6(0.029) = The expected size of a randomly selected household is about 2.4 people. 2. ANS: In this situation, n = 5, p = 0.25, and q = a) Include Number Correct, x P(x) b) Add the probabilities of having exactly 3, 4, and 5 successes. P(passing) = = The probability of passing is about c) E(X) = 5(0.25) = 1.25 The expectation of 1.25 is the same using either method. Any differences are due to rounding. 3. ANS: For this hypergeometric distribution, n = 40 and r = 7. a) In this case, a = 16, and x = 7. The probability that the winners are all children is about b) In this case, a = 16, and x = 2. The probability that exactly two of the winners are children is about c)

13 The probability that exactly three of the winners are children and two are women is about d) The expected number of men, women, and children are 1.75, 2.45, and 2.8, respectively.

Probability Distributions

4.1 Probability Distributions Random Variables A random variable x represents a numerical value associated with each outcome of a probability distribution. A random variable is discrete if it has a finite