1. (Binomial)

A sharpshooter is taking aim at a target. It is known that the sharpshooter hits the bullseye 80% of the time. If the sharpshooter took 20 consecutive and independent shots at the target, determine the following:

a) What is the probability the sharpshooter hits exactly 17 out of 20 shots?

b) What is the probability the sharpshooter hits at most 13 shots?

c) What is the probability the sharpshooter hits at least 8 shots?

d) What is the probability the sharpshooter hits between 7 and 15 shots?

e) What is the expected number of shots the sharpshooter will hit? What is the variance and standard deviation?

In binomial distribution, to calculate probabilities we use combination formula and multiply by probabilities of success and failure. Also, expected value E(X) is np, variance VAR(X) is np(1-p) and standard deviation SD(X) is root of VAR(X).

Explanation:

This question pertains to the use of

binomial distribution

in mathematics. Binomial distribution is a type of probability distribution that is used when there are exactly two mutually exclusive outcomes of a trial, often referred to as success and failure. The sharpshooter's shots can be considered as a series of Bernoulli trials, where each shot is independent of the others and each has the same probability of hitting the bullseye. To calculate the binomial probability, we use the formula:

P(X = k) = C(n, k) * (p^k) * ((1 - p)^(n-k))

. The expected value E(X) of a binomial distributed random variable X = number of successes is np and the variance VAR(X) is np(1-p), the standard deviation SD(X) is the square root of the variance. Here, 'n' is the number of trials (20 shots), 'p' is the probability of success (80% or 0.8), and 'k' is the number of successes. Using this formula a) for 17 shots, b) sum up probabilities for 0 to 13 shots, c) sum up probabilities for 8 to 20 shots d) sum up probabilities for 7 to 15 shots e) Expected number = 20*0.8 = 16 shots, variance = 20*0.8*0.2 = 3.2 shots, standard deviation = sqrt(3.2)= ~1.79 shots.

Learn more about Binomial Distribution here:

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