Question

What distribution best models each situation? (No actual work, just which distribution best models the problem)

A. According to the Daily Racing Form, the probability is about 0.67 that the favorite in a horse race will finish in the money (first, second or third place). Suppose that you always bet the favorite “across the board”, which means that you win something if the favorite finishes in the money. What probability distribution describes the number of races that you bet until you win something three times?

Hypergeometric/ Geometric/ Negative Binomial/ Binomial/ Poisson/ or None of these?

B. Astronomers estimate that as many as 100 billion stars in the Milky Way galaxy are encircled by planets. If so, we may have a plethora of cosmic neighbors. Let p denote the probability that any such solar system contains intelligent life. You would like to estimate, how small can p be and still give a 50-50 chance that we are not alone. What probability distribution will be used?

Hypergeometric/ Geometric/ Negative Binomial/ Binomial/ Poisson/ or None of these?

C. Because of her past convictions for mail fraud and forgery, Jody has a 30% chance each year of having het tax returns audited. What probability distributions will be used to find the chance that she will escape detection for next three years, but will be audited after? (Assume that she exaggerates, distorts, misrepresents, lies, and cheats every year)

Hypergeometric/ Geometric/ Negative Binomial/ Binomial/ Poisson/ or None of these?

D. According to the Daily Racing Form, the probability is about 0.67 that the favorite in a horse race will finish in the money (first, second or third place). What probability distribution describes the number of times the favorite finishes in the money in the next five races?

Hypergeometric/ Geometric/ Negative Binomial/ Binomial/ Poisson/ or None of these?

E. Recently married, a young couple plans to continue having children until they have two boys. What probability distribution you will use to estimate the couple’s expected family size?

Hypergeometric/ Geometric/ Negative Binomial/ Binomial/ Poisson/ or None of these?

Answer 1

(a)

Here we need three success with probability of success each time is p = 0.67.

It is negative binomial distribution with parameters r=3 and p=0.67.

(b)

Here Possion distribution will be used.

(c)

Here we need to find the probability until we get first success. The probability of success in any trial is 0.30.

It is geometric distribution with parameter p = 0.30.

(d)

Since each race is independent from other. Each time probability of winning is 0.67. The number of trial is 5.

It is binomial distribution.

(e)

Each time probability of baby boy is P(boy) = 0.5.

Family will continue having children until they have two boys (that is two success). So it is negative binomial distribution with parameters r=2 and p=0.5.