Problem: Susan is taking Western Civilization this semester on a pass/fail basis. The department teaching the course has a history of passing 67% of the students in Western Civilization each term. Let n = 1, 2, 3, ... represent the number of times a student takes Western Civilization until the first passing grade is received. (Assume the trials are

independent.)

1. Write out a formula for the probability distribution of the random variable n. (Use p and n in your answer.)
2. What is the probability that Susan passes on the first try (n = 1)? (Use 2 decimal places.)
3. What is the probability that Susan first passes on the second try (n = 2)? (Use 3 decimal places.)
4. What is the probability that Susan needs three or more tries to pass Western Civilization? (Use 3 decimal places.)
5. What is the expected number of attempts at Western Civilization Susan must make to have her (first) pass? Hint: Use μ for the geometric distribution and round.

Problem: Bob is a recent law school graduate who intends to take the state bar exam. According to the National Conference on Bar Examiners, about 56% of all people who take the state bar exam pass. Let n = 1, 2, 3, … represent the number of times a person takes the bar exam until the first pass.

1. Write out a formula for the probability distribution of the random
   variable n. (Use p and n in your answer.)
2. What is the probability that Bob first passes the bar exam on the second try (n = 2)? (Use 3 decimal places.)
3. What is the probability that Bob needs three attempts to pass the bar exam? (Use 3 decimal places.)
4. What is the probability that Bob needs more than three attempts to pass the bar exam? (Use 3 decimal places.)
5. What is the expected number of attempts at the state bar exam Bob must make for his (first) pass? Hint: Use μ for the geometric distribution and round.

Problem: USA Today reported that the U.S. (annual) birth rate is about 21 per 1000 people, and the death rate is about 10 per 1000 people.

1. Explain why the Poisson probability distribution would be a good choice for the random variable r = number of births (or deaths) for a community of a given population size?
   Frequency of births (or deaths) is a common occurrence. It is reasonable to assume the events are independent.
   Frequency of births (or deaths) is a common occurrence. It is reasonable to assume the events are dependent.
   Frequency of births (or deaths) is a rare occurrence. It is reasonable to assume the events are dependent.
   Frequency of births (or deaths) is a rare occurrence. It is reasonable to assume the events are independent.
2. In a community of 1000 people, what is the (annual) probability of 10 births? What is the probability of 10 deaths? What is the probability of 16 births? 16 deaths? (Use 4 decimal places.)
   P(8 births) =
   P(8 deaths) =
   P(15 births) =
   P(15 deaths) =
3. Repeat part (b) for a community of 1500 people. You will need to use a calculator to compute P(10 births) and P(16 births). (Use 4 decimal places.)
   P(8 births) =
   P(8 deaths) =
   P(15 births) =
   P(15 deaths) =
4. Repeat part (b) for a community of 750 people. (Use 4 decimal places.)

P(8 births) =

P(8 deaths) =

P(15 births) =

P(15 deaths) =

Much of Trail Ridge Road in Rocky Mountain National Park is over 12,000 feet high. Although it is a beautiful drive in summer months, in winter the road is closed because of severe weather conditions. Sustained gale-force winds (over 32 miles per hour and often over 90 miles per hour) occur on the average of 1.0 times every 63 hours at a Trail Ridge Road weather station.

1. Let r = frequency with which gale-force winds occur in a given time interval. Explain why the Poisson probability distribution would be a good choice for the random variable r.
   Frequency of gale-force winds is a common occurrence. It is reasonable to assume the events are dependent.
   Frequency of gale-force winds is a common occurrence. It is reasonable to assume the events are independent.
   Frequency of gale-force winds is a rare occurrence. It is reasonable to assume the events are dependent.
   Frequency of gale-force winds is a rare occurrence. It is reasonable to assume the events are independent.
2. For an interval of 119 hours, what are the probabilities that r = 2, 3, and 4? What is the probability that r < 2? (Use 2 decimal places for λ. Use 4 decimal places for your answers.)
   P(2) =
   P(3) = 0
   P(4) =
   P(r < 2) =
3. For an interval of 188 hours, what are the probabilities that r = 3, 4, and 5? What is the probability that r < 3? (Use 2 decimal places for λ. Use 4 decimal places for your answers.)

P(3) =

P(4) =

P(5) =

P(r < 3) =

Chances: Risk and Odds in Everyday Life, by James Burke, reports that only 2.9% of all local franchises are business failures. A Colorado Springs shopping complex has 137 franchises (restaurants, print shops, convenience stores, hair salons, etc.).

1. Let r be the number of these franchises that are business failures. Explain why a Poisson approximation to the binomial would be appropriate for the random variable r.
   The Poisson approximation is good because n is large, p is small, and np > 10.
   The Poisson approximation is good because n is large, p is large, and np < 10.
   The Poisson approximation is good because n is large, p is small, and np < 10.
   The Poisson approximation is good because n is small, p is small, and np < 10.
   What is n? What is p? What is λ (rounded to the nearest tenth)?
   n =
   p =
   λ =
2. What is the probability that none of the franchises will be a business failure? (Use 4 decimal places.)
3. What is the probability that two or more franchises will be a business failure? (Use 4 decimal places.)
4. What is the probability that four or more franchises will be a business failure? (Use 4 decimal places.)

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