Question

Stephanie Kerry is a very good basketball player. Each time she attempts a free throw, she misses it with a probability of only 5% (independently of other free throw attempts). This month, she will attempt 1000 free throws.

(a) What is the distribution of X, the total number of free throws that Stephanie misses this month? Give its name and compute its parameters.

(b) What is the probability that Stephanie will miss at least 61 free throws this month?

(c) Use a Poisson approximation to give an approximation for the probability that Stephanie will miss at least 61 free throws this month.

(d) By the end of the 15th day of the month, Stephanie has already missed 55 free throws. Given this information, what is the chance that she will miss at most 60 free throws in total this month? Use the Poisson approximation in answering this question.

(e) At the end of the month, Stephanie will look at her total number of missed free throws, X. If X ≥ 50, she will put 5 dollars in a jar for each free throw she misses. For example, she puts 250 dollars in the jar if X = 50. If X < 50, she will leave the jar empty. What is the expected number of dollars in the jar? Use the Poisson approximation in answering this question.

Answer 1

(a)

The distribution of X is Binomial distribution with n = 1000 and p = 0.05.

X ~ Binomial(n = 1000, 0.05)

(b)

Probability that Stephanie will miss at least 61 free throws this month = P(X 61)

Using binomial distribution function in R studio, pbinom(60, 1000, 0.05, lower.tail = FALSE) is 0.06706252

(c)

Using Poisson approximation, X will follow Poisson distribution with = 1000 * 0.05 = 50

Probability that Stephanie will miss at least 61 free throws this month = P(X 61) = 0.07216018

Using function, ppois(60, 50, lower.tail = FALSE) in R Studio.

(d)

Rate for 15 days, = 50 / 2 = 25 per 15 days

By the end of the 15th day of the month, Stephanie has already missed 55 free throws.

Probability that that she will miss at most 60 free throws in total this month = Probability that that she will miss at most 5 free throws in next 15 days = P(X 5) = 0.000001397112

(Using the command, ppois(5, 25) in R Studio)