bus stats

Section 4.1

1. An experiment has three steps with three outcomes possible for the first step, two outcomes possible for the second step, and four outcomes possible for the third step. How many experimental outcomes exist of the entire experiment?

2. How many was can three items be selected from a group of six items? Use the letters A, B, C, D, E, and F to identify the items, and list each different combinations of three items.

3. How many permuations of three items can be selected from a group of 6? Use the letters A, B, C, D, E, and F to identify the items, and list each of the permutations of items B, D, and F only.

4. Consider the experiment of tossing a coin three times.

a) Develop a tree diagram for the experiment

b) List the sample space(the collection of all outcomes)

c) What is the probability for each experimental outcome?

5. Suppose an experiment has five equally likely events: E1, E2, E3, E4, E5. Since these events are equally likely, what is the probability of each event? Show that the sum of the probabilities of all the events is 1.

6. An experiment with three outcomes: A, B, and C has been repeated 50 times. It was learned that A occurred 20 times, B occurred 13 times, and C occurred 17 times. Assign probabilities to the outcomes. What method did you use?

7. A decision maker subjectively assigned the following probabilities to the four outcomes of an experiment:

P(E1) = 0.1 P(E2) = 0.15 P(E3) = 0.40 and P(E4) = 0.20

Are these probability assignments valid? Explain. (Hint: when assigning probabilities to outcomes of a random experiment, what two criteria must be satisfied?)

8. In the city of Milford, applications for zoning changes go through a two-step process: a review by the planning commission and a final decision by the city council. At step 1 the planning commission reviews the zoning change request and makes a positive or negative recommendation. At step 2 the city council then votes to approve or disapprove the zoning change. Consider the application process as an experiment.

a. List the sample space for this experiment.

b. Construct a tree diagram for the experiment.

9. Suppose that a bank has 50 accounts and randomly selects 4 of them for screening. How many different samples of four accounts are possible?

10. The table below shows the percentages of students who graduate with debt as well as the average debt.

a. If you randomly choose a graduate of Morehouse College, what is the probability that this individual graduated with debt?

b. If you randomly chose one of these eight institutions for a follow-up study, what is the probability that you will chose an institution with more than 60% of its graduates having debt?

c. If you randomly chose one of these eight institutions for a follow-up study, what is the probability that you will chose an institution whose graduates have an average debt of more than $30,000?

d. What is the probability that a Pace University graduate does not have debt?

11. A survey of motorcycle helmets was conducted and the following information was gathered:

Helmet Type

a) Use the sample data to compute an estimate of the probability that a motorcyclist wears a DOT-compliant helmet.

b) What is the probability of DOT-compliant helmet use by region of the country? What region has the highest probability of DOT-compliant helmet use?

12. The Powerball lottery is a game where a participant selects five numbers 1 through 55 and then a Powerball number from 1 through 42. To determine the winning numbers for each game, lottery officials draw five white balls out of a drum with 55 white balls, and one red ball our of a drum of 42 red balls. To win the jackpot, a participant’s numbers must match the numbers on the five white balls in any order and the number on the red Powerball.

a) Compute the number of ways the first five numbers can be selected.

b) Compute the number of ways to choose the first five numbers and powerball.

c) What is the probability of winning the Powerball jackpot?

Section 4.2

13. An experiment has four equally likely outcomes: E1, E2, E3, E4

a) What is the probability that E2 occurs?

b) What is the probability that any two of the outcomes occur (e.g. E1 or E3)?

c) What is the probability that any three outcomes occur? (E1, E2, or E4)?

14. Consider the experiment of selecting a playing card from a deck of 52 playing cards. Each card corresponds to a sample point with a 1/52 probability.

a) List the sample points in the event an ace is selected.

b) List the sample points in the event a club is selected.

c) List the sample points in the event a face card is selected(Jack, Queen, or King).

d) Find the probabilities associated with each of the events in parts a-c.

15. Consider the experiment of rolling a pair of dice. Suppose that we are interested in the sum of the face values showing on the dice.

a) How many sample points are possible? (Hint: Use the counting rule for multiple-step experiments)

b) List the sample space(all of the sample points).

c) What is the probability of obtaining a value of 7?

d) What is the probability of obtaining a value of 9 or greater?

e) Because each roll has six possible even values (2, 4, 6, 8, 10, or 12) and only five possible odd values (3, 5, 7, 8, and 11), the dice show even values more often that odd values. Do you agree with this statement? Explain.

f) What method did you use to assign the probabilities requested?

16) To investigate how often families eat at home, the following survey results were collected from 496 families:

a) The probability the family eats no meals at home during the week.

b) The probability the family eats at least four meals at home during the week.

c) The probability the family eats two or fewer meals at home during the week.

17) Do you think the government protects investors adequately? This questions was asked to investors in both the US and Great Britain. The responses are summarized in the table below:

a) Estimate the probability that an investor in the United States thinks the government is not protecting investors adequately.

b) Estimate the probability that an investor in Great Britain thinks the government is not protecting investors adequately or is unsure the government is protecting investors adequately.

c) For a randomly selected investor form these two countries, estimate the probability that the investor thinks the government is not protecting investors adequately.

Section 4.3

18) Suppose that we have a sample space with five equally likely outcomes: : E1, E2, E3, E4, E5

Let A = {E1, E2}

B = {E3, E4}

C = {E2, E3, E5}

a) Find P(A), P(B), and P(C)

b) Find P(AB). Are A and B mutually exclusive? Explain.

c) Find Ac, Cc , P(Ac), and P(Cc)

d) Find ABc and P(ABc)

e) Find P(BC)

19) Suppose that we have a sample space S = {E1, E2, E3, E4, E5, E6, E7}. The following probability assignments apply:

P(E1) = 0.05, P(E2) = 0.20, P(E3) = 0.20, P(E4) = 0.25, P(E5) = 0.15, P(E6) = 0.10, P(E7) = 0.05

Let A = {E1, E4, E6}

B = {E2, E4, E7}

C = {E2, E3, E5, E7}

a) Find P(A), P(B), and P(C)

b) Find AB and P(AB)

c) Find AB and P(AB)

d) Are events A and C mutually exclusive? Explain.

e) Find Bc and P(Bc)

20) Clarkson University surveyed alumni to learn more about what they think of the college. The responses where either that the college fell short, met expectations, or surpassed expectations. The results show that 4% did not provide a response, 26% said that their experience fell short of expectations, and 65% of the respondents said that their experience met expectations.

a) If we chose an alumnus at random, what is the probability that the alumnus would say their experience surpassed expectations?

b) If we chose an alumnus at random, what is the probability that the alumnus would say their experience met or surpassed expectations?

22) A survey of magazine subscribers showed that 45.8% rented a car during the past 12 months for business reasons, 54% rented a car during the past 12 months for personal reasons, and 30% rented a car during the past 12 months for both business and personal reasons.

a) What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons?

b) What is the probability that a subscriber did not rent a car duing the past 12 months for either business or personal reasons?