Question

In: Statistics and Probability

Stephanie Kerry is a very good basketball player. Each time she attempts a free throw, she...

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.

Expert Solution

(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)

orchestra answered 2 years ago

A basketball player makes 64% of her free throw attempts. Suppose she is awarded 3 free...

A basketball player makes 64% of her free throw attempts. Suppose she is awarded 3 free throws during a play, and let X be the number of free throws she makes during this trial. Complete the following probability distribution table. Round the probabilities to 3 decimals.

A basketball player completes a free throw 80% of the time. In practice the player goes...

A basketball player completes a free throw 80% of the time. In practice the player goes to the free throw line and takes 5 shots in a row. a) Make a table showing the probability distribution of successes and their probabilities b) Draw the probability histogram. c) What is the shape of the distribution? d) What is the mean? e) What is the standard deviation?

A basketball player shoots 3 free throws. On each throw, she has a 70% chance of...

A basketball player shoots 3 free throws. On each throw, she has a 70% chance of making the shot. Assume the outcomes of the throws are independent. What is the probability that she makes the first two free throws and then misses the last one?

A basketball player has a 50 % chance of making each free throw. What is the...

A basketball player has a 50 % chance of making each free throw. What is the probability that the player makes at most eight out of ten free throws?

On a womens basketball team, one player can make 61% of free throws she attempts. A...

On a womens basketball team, one player can make 61% of free throws she attempts. A success is making a free throw (in basketball, "making a free throw" means the ball successfully went through the hoop.) The random variable X = the number of free throws made out of the seven attempted. Assume that outcomes of free throw attempts can be considered independent of each other. (a) What is the probability that in a given game, she attempts 7 free...

To test his free throw skills, a basketball player shoots 200free throw shots in a...

To test his free throw skills, a basketball player shoots 200 free throw shots in a row. He makes 171 of them. Based on this, what is the probability he will make his first free throw in his next game? What method of calculating probability did you use to calculate this?

2. A basketball player hits 80% of her free-throws. If she gets four (4) attempts, with...

2. A basketball player hits 80% of her free-throws. If she gets four (4) attempts, with what probability will she: a. hit exactly three free-throws; b. three or more free-throws; c. one or more free-throws? d. What is the expected number of free-throws she will hit? (The mean).

A basketball player makes 75% of the free throws he tries. If the player attempts 10...

A basketball player makes 75% of the free throws he tries. If the player attempts 10 free throws in a game, find the probability that: the player will make at most six free throws. the player will make at least eight free throws.

A basketball player was an 84% free throw shooter. a. At the moment you turn the...

A basketball player was an 84% free throw shooter. a. At the moment you turn the game on he is 5 of 7 shooting from the free-throw line. What is the probability that he made 5 of his first 7 shots? b. What is the probability that he made his 5th shot on his 7th attempt? c. What is the probability that he made his first shot on his third attempt?

It is known that a certain basketball player will successfully make a free throw 87.4% of...

It is known that a certain basketball player will successfully make a free throw 87.4% of the time. Suppose that the basketball player attempts to make 14 free throws. What is the probability that the basketball player will make at least 11 free throws? Let XX be the random variable which denotes the number of free throws that are made by the basketball player. Find the expected value and standard deviation of the random variable. E(X)= σ= Suppose...