We appreciate your visit to 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. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
Final answer:
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.
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