# A dodecahedral die has 12 faces, numbered 1-12.

1.A dodecahedral die has 12 faces, numbered 1-12. If the die is weighted in such a way that 2 is twice as likely to land facing up as 1, 3 is three times as likely to land facing up as 1, and so on, what is the probability distribution for the face landing up?

Outcome123456789101112

Probability

2.Use the given information to find the indicated probability.

A and B are mutually exclusive.

P(A) = .3, P(B) = .1. Find P((A ∪ B)’).

P((A ∪ B)’) =

3.Lance the Wizard has been informed that tomorrow there will be a 40% chance of encountering the evil Myrmidons and a 20% chance of meeting up with the dreadful Balrog. Moreover, Hugo the Elf has predicted that there is a 10% chance of encountering both tomorrow. What is the probability that Lance will be lucky tomorrow and encounter neither the Myrmidons nor the Balrog?

4.Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, four green ones, two white ones, and three purple ones. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.]

She has all the red ones.

5.The Sad State Lottery requires you to select a sequence of four different numbers from 0 through 60. (Order is important.) You are a Winner if your sequence agrees with that in the drawing, and you are a Booby Prize Winner if your selection of numbers is correct, but in the wrong order. (Enter your answers as exact answers.) What is the probability of being a Winner?

What is the probability of being a Booby Prize Winner?

What is the probability that you are either a Winner or a Booby Prize Winner?

6.Compute the indicated quantity.

P(A | B) = .1, P(B) = .7. Find P(A ∩ B).

P(A ∩ B) =

7.Compute the indicated quantity.

P(A) = .2, P(B) = .3. A and B are independent. Find P(A ∩ B).

P(A ∩ B) =

8.According to a certain news poll, 74% agreed that it should be the government’s responsibility to provide a decent standard of living for the elderly, and 50% agreed that it would be a good idea to invest part of their Social Security taxes on their own. If agreement with one of these propositions is independent of agreement with the other, what is the probability that a person agreed with both propositions? (Round your answer to two decimal places.) HINT [See Quick Examples on Independence.]

9.Use Bayes’ theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places. HINT [See Example 3.]

P(A | B) = .6, P(B) = .2, P(A | B’) = .9. Find P(B | A).

P(B | A) =

10.Suppose that it rains in Spain an average of once every 15 days, and when it does, hurricanes have a 2% chance of happening in Hartford. When it does not rain in Spain, hurricanes have a 1% chance of happening in Hartford. What is the probability that it rains in Spain when hurricanes happen in Hartford?

11.In a large on-the-job training program, half of the participants are female and half are male. In a random sample of three participants, what is the probability that an investigator will draw at least two males?† (Round your answer to four decimal places.)

P(X ≥ 2) =

12.The probability that a randomly selected teenager watched a rented video at least once during a week was 0.74. What is the probability that at least 3 teenagers in a group of 5 watched a rented movie at least once last week? (Round your answer to four decimal places.)

P(X ≥ 3) =

13.Assume that on a standardized test of 100 questions, a person has a probability of 85% of answering any particular question correctly. Find the probability of answering between 80 and 90 questions, inclusive, correctly. (Assume independence, and round your answer to four decimal places.)

P(80 ≤ X ≤ 90) =

14.Calculate the standard deviation σ of X for the probability distribution. (Round your answer to two decimal places.)

σ =

x−5 −10 2 5 10

P(X = x) 0.30.20.10.10.30

15.Complete the following probability distribution table and then calculate the stated probabilities. HINT [See Quick Example 5.]

Outcomeabcde

Probability 0.1 0.08 0.4 0.02?

(a) P({a, c, e})

P({a, c, e}) =

(b) P(E ∪ F), where E = {a, c, e} and F = {b, c, e}

P(E ∪ F) =

(c) P(E’), where E is as in part (b)

P(E’) =

(d) P(E ∩ F ), where E and F are as in part (b)

P(E ∩ F) =

16.According to an article, 17.7% of Internet stocks that entered the market in 1999 ended up trading below their initial offering prices. If you were an investor who purchased three Internet stocks at their initial offering prices, what was the probability that at least two of them would end up trading at or above their initial offering price? (Round your answer to four decimal places.)

P(X ≥ 2) =

17.Z is the standard normal distribution. Find the indicated probability. HINT [See Example 1.] (Round your answer to four decimal places.)

P(−0.7 ≤ Z ≤ 1.24)

=

18.Find the probability that a normal variable takes on values more than 2/3 standard deviations away from its mean. (Round your answer to four decimal places.)

19.If you roll a die 200 times, what is the approximate probability that you will roll fewer than 25 ones, inclusive? (Round your answer to two decimal places.)

20.This exercise uses the normal probability density function and requires the use of either technology or a table of values of the standard normal distribution.

The cash operating expenses of the regional phone companies during the first half of 1994 were distributed about a mean of $29.8 per access line per month, with a standard deviation of $2.25. Company A’s operating expenses were $28.00 per access line per month. Assuming a normal distribution of operating expenses, estimate the percentage of regional phone companies whose operating expenses were closer to the mean than the operating expenses of Company A were to the mean. (Round your answer to two decimal places.)