Unit 6 Practice Questions 1. For each of the following situations, determine the complete sample space of outcomes. (a) A student who waits for two buses every day records the status of the arrival of each bus. The bus is either late, on time, or early. The outcome of interest is the status of each bus for one day. (b) Two standard decks of cards (consisting of 13 hearts, 13 diamonds, 13 spades and 13 clubs) is shuffled. One card is randomly selected from each deck. The outcome of interest is the suit that is selected from each deck. (c) There are three coins in a hat – one gold, one silver and one copper. You will randomly select coins without replacement until the gold coin is selected. The outcome of interest is the colour of each coin that is selected during this process. 2. The manager of a gas station and convenience store tabulates the following facts: • 45% of customers purchase gasoline (G). • 27% of customers purchase cigarettes (C). • 67% of customers purchase gasoline or a drink (D). • 14% of customers purchase gasoline and cigarettes. • 18% of customers purchase a gasoline and a drink. • 9% of customers purchase cigarettes and a drink. • 5% of customers purchase all three items. (a) The outcome of interest is which, if any, of the items a customer purchases. Give the appropriate sample space. (b) What is the probability that a randomly selected customer purchases gasoline or cigarettes? (c) What is the probability that a randomly selected customer purchases a drink? (d) Are any of the pairs of events independent of one another? Explain. (e) Are any of the pairs of events mutually exclusive? Explain. (f) What is P (C ∩ Dc ), i.e., the probability that a randomly selected customer purchases cigarettes but not a drink? (g) What is the probability that a customer purchases none one of the three items? 3. Suppose we have two bowls full of candies. Each bowl contains four different flavours of candy – grape (which are purple), lemon (which are yellow), cherry (which are red) and raspberry (which are also red). (a) We will randomly select one candy from each bowl. The outcome of interest is the flavour of each of the two candies. Write out the complete sample space of outcomes. (b) Suppose instead that we randomly select one candy from each bowl, and the outcome of interest is the colour of each of the two selected candies. Write out the complete sample space of outcomes. Now suppose: • Bowl # 1 contains 2 grape candies, 7 lemon candies, 8 cherry candies and 3 raspberry candies. • Bowl # 2 contains 6 grape candies, 5 lemon candies, 2 cherry candies and 7 raspberry candies. (c) What is the probability that the two selected candies are the same flavour? (d) What is the probability that the two selected candies are different colours? (e) What is the probability that the first selected candy is lemon or that the second selected candy is cherry? (f) Let X be the number of grape candies that are selected. Find the probability distribution of X. 4. Eugenie Bouchard is playing Serena Williams in a best-of-three match in the final of a Grand Slam tennis tournament. (The first player to win two sets is the winner.) Being the higher ranked player, suppose that Williams has a 60% chance of winning any given set and that the outcome of one set is independent of any other. (a) The outcome of interest is the winner of each set that is played. List the complete sample space of possible outcomes. (b) What is the probability that the match lasts three sets? (c) What is the probability that Bouchard wins the match? 5. We have two boxes, each containing several numbered balls: • Box 1 contains five balls – two with the number 0 and three with the number 1. • Box 2 contains four balls – one with the number 0, two with the number 1 and one with the number 2. You will randomly select one ball from each box. (a) The outcome of interest is the number on each of the two selected balls. List the complete sample space of outcomes and calculate the probability of each outcome. (b) Let X be the sum of the numbers on the two selected balls. Find the probability distribution of X. (c) Let Y be the maximum of the numbers on the two selected balls. Find the probability distribution of Y . 6. A student must pass through two sets of traffic lights on his way to university each day. The first light is green 35% of the time, yellow 8% of the time and red 57% of the time. The second light is green 42% of the time, yellow 11% of the time and red 47% of the time. Suppose that it is known that the two sets of lights operate independently. (a) On any given trip to university, what is the probability that both lights encountered by the student will be the same colour? (b) What is the probability that the first light will be green or that the second light will be red? (c) Let X be the number of times the student stops at a light on his way to school. (He will only stop if the light is red, and not if it is yellow.) Find the probability distribution of X. 7. Another student must pass through 12 sets of traffic lights on his way to university each day. Suppose that each of the lights is green 36% of the time, yellow 5% of the time, and red 59% of the time. Suppose it is known that the traffic lights function independently. (a) What is the probability that the student encounters exactly five red lights on his way to university one day? (b) What is the probability that the student encounters at least three green lights on his way to university one day? (c) In a five-day school week, what is the probability that the student encounters exactly seven yellow lights on his way to university? 8. Determine whether the variable X has a binomial distribution in each of the following cases. If it does, determine the values of the parameters n and p and calculate P (X = 3). If it doesn’t, explain why not. (a) You randomly select one card from each of eight different decks of playing cards. X = number of clubs selected (b) You are dealt a five-card poker hand. X = number of diamonds in your hand. (c) You toss five fair coins – a toonie, a loonie, a quarter, a dime and a nickel. X = number of coins that land on Heads (d) Suppose that 10% of all McDonald’s customers buy a Big Mac. You will record the orders of the next 40 customers. X = number of customers who buy a Big Mac. (e) A fair die is rolled 20 times. X = number of times the number 6 is observed. (f) A fair die is repeatedly rolled. X = number of rolls required to observe the number 5 for the third time. (g) An airplane carrying 100 passengers has two emergency exits – one at the front of the plane and one over the wings. The plane makes an emergency landing. X = number of passengers that exit the plane through the front emergency exit. (h) A boy asks the same girl to go on a date every day for a week. X = number of times the girl agrees to go on a date with the boy 9. m&m’s chocolate candies come in six different colours. Suppose the colour distribution is as follows: Blue Brown Green Orange Red Yellow Colour Probability 0.12 0.23 0.17 0.16 0.22 0.10 (a) If you randomly select 10 m&m’s, what is the probability that 2 of them are blue? (b) If you randomly select 25 m&m’s, what is the probability that 7 of them are brown? (c) If you randomly select 18 m&m’s, what is the probability that 4 of them are green? (d) If you randomly select 15 m&m’s, what is the probability that less than 3 of them are orange? (e) If you randomly select 7 m&m’s, what is the probability that at least 2 of them are red? (f) If you randomly select 20 m&m’s, what is the probability that more than 2 of them are yellow?
