#### Stat 414 Midterm Exam #2 Review Sheet

For each of the following situations, identify whether X is:

• uniform
• hypergeometric
• binomial
• geometric
• negative binomial
• Poisson

1. A grocery store has 10 checkout lanes. During a busy hour the probability that any given lane is occupied is 0.75. Assume that the lanes are occupied or not occupied independently of eachother. Find the probability that at most 5 lanes are occupied.

2. An office supply warehouse receives an order for 3 computers. The warehouse has 50 computers in stock, two of which are defective. The order is filled randomly drawing from the computers in stock. What is the probability that a customer is not sent any of the defective computers?

3. A typist makes an average of 1 typographical error every 20 pages. What is the probability that the typist makes at most 3 errors in 20 pages?

4. A couple decides to have children until they have one female. What is the probability that the couple ends up with 4 children?

5. If a couple decides to have children until they have two females, what is the probability that the could ends up with 6 children?

6. A roulette wheel is divided into 25 sectors of equal area numbered from 1 to 25. What is the p.d.f. of X, the number that occurs when the wheel is spun?

7. The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive.

8. A random committee of size 3 is selected from 4 doctors and 2 nurses. What is the probability that there is at most 1 nurse on the committee?

9. The probability that a person living in a certain city owns a dog is estimated to be 0.3. Find the probability that the tenth person randomly interviewed in this city is the fifth one to own a dog.

10. Suppose the probability is 0.8 that any given person will believe a tale about the transgressions of a famous actress. What is the probability that the third person to hear this tale is the first one to believe it?

11. On the average a certain intersection results in 3 accidents per month. What is the probability that in any given month at this intersection less than 3 accidents occur?

#### Solutions

1. If X is number occupied, X is binomial with n = 10 and p = 0.75. If Y is number unoccupied, Y is Binomial with n=10 and p = 0.25. The X = at most 5 is Y = at least 5. P(Y at least 5) = 1 - P(Y <= 4) = 1 - 0.9219 = 0.0781.

2. X is hypergeometric. P(X = 0) = [(2 choose 0) x (48 choose 3)]/(50 choose 3)

3. Poisson with l=1. P(X < = 3) = 0.981

4. Geometric. P(fourth try is first success) = 1/16

5. Negative binomial. P(six tries to get 2 successes) = (5 choose 1)x(1/2)^6 = 0.078

6. f(x) = 1/25, x = 1, 2, ...., 25.

7. Binomial with n = 7 and p = 0.9. P(X = 5) = 0.124

8. Hypergeometric. P(X<=1)
= [(2 choose 0) x (4 choose 3)]/(6 choose 3) + [(2 choose 1) x (4 choose 2)]/(6 choose 3)

9. Negative binomial. P(X=10) = (9 choose 4)x(0.3)^5x(0.7)^5 = 0.051

10. Geometric. P(X=3) = 0.2^2(0.8) = 0.032

11. Poisson with l=3. P(X < 3) = P(X<=2) = 0.423.