AP Stats: Chapter 8 Review
1. Describe the difference between a geometric and binomial setting and give an example of each.
Binomial: Variable X is the number of successes in a set number of observations
Geometric: Variable X is the number of trials it takes to find the first success
2. Based on a recent survey of government workers, it is estimated that about 85 percent have had some
college education. If 6 government workers are selected at random what is the probability that:
a. At least 5 have had some college education.
P(X>5) = .7765
b. At least 2 and no more that four have had some college education.
P(2≤X≤4) = .22310
c. Four have not had some college education.
P(X=2) = .0055
d. What is the expected number of government workers who have had some college education?
μ=np = 5.1  6
3. You are participating in a “question bee” in history class. You remain in the game until you give
your first incorrect answer. The questions are all multiple choice with four possible answers, exactly
one of which is correct. Unfortunately, you have not studied for this bee and you simply guess
randomly on each question.
a. What is the probability that you are still in the bee after the first round?
P(X>1) = .25
b. What is the probability you are still in the bee after the third round?
P(X>3) = .0156
c. What is the expected number of rounds you will be in the bee?
μ = 1/.75 = 1.33  2
d. If an entire class of 32 is simply guessing on each question, how many do you expect to be
left after the third round?
Binomial: n = 32 p = .25
After 1st round 8 stay (np = 8), After 2nd round (np=2 if n=8), After 3rd round (np=.51 if n=2)
4. Suppose 12% of the engines manufactured on a certain assembly line have at least one defect.
Engines are randomly sampled from this line one at a time and tested.
a. What is the expected number of engines that need to be tested before finding the first engine
without a defect? Before finding the third?
μ=1/.88 = 1.142 (For the third μ = 1.14*3 = 3.424)
b. What is the standard deviation of the number of engines that need to be tested before finding
the first engine without a defect?
σ = √(.12/.88^2) = .3936
c. If it costs $100 to test one engine, what are the expected value and standard deviation of the
cost of inspection up to and including the first engine without a defect?
μ= $114 σ = $39.36 (use rules of means and variances)
d. Will the cost of inspection often exceed $200? Explain.
z = 2.18 so P(C > $200) = .014 (Rarely)
5. One prison experiences a 70% return rate for released prisoners.
a. Find the probability of getting 12 returning prisoners from the next group of 15 that are
released.
P(X = 12) = .17004
b. What is the average number of prisoners returning out of 15?
μ = np = 10.5  11 prisoners
6. An oil exploration firm is to drill wells at a particular site until it finds one that will produce oil.
Each well has a probability of .1 of producing oil. It costs the firm $50,000 to drill each well.
a. What is the expected number of wells to be drilled?
μ = 1/.1 = 10 wells
b. What re the expected value and standard deviation of the cost of drilling to get the first
successful well?
μ = $50,000(10) = $500,000
σ = $474,341.65
c. What is the probability that it will take at least 5 tries to get the first successful well? At least
15?
P(X>4) = .6561 P(X>14) = .2288
d. Find the probability it takes more than 10 tries to find the first successful well.
P(X >10) = .3487
7. 12.7% of Americans live below the poverty level. Suppose you plan to sample at random 100
Americans and count the number who live below the poverty level. What is the probability you
count 10 or fewer?
P(X≤10) .2610
8. What rules should you check before assuming a binomial distribution is approximately normal?
np≥10 AND n(1-p) ≥ 10
9. How do you find the z-score?
z = x – μ When the distribution is (or is assumed) approximately normal.
σ
10. For a random variable X that takes on an approximately normal distribution, what is the probability
X will fall within 2 standard deviations from the mean?
95% = P(-2 < z < 2)