Question

# A random variable follows a binomial distribution with a probability of success equal to 0.52. For...

A random variable follows a binomial distribution with a probability of success equal to 0.52. For a sample size of n=7, find the values below.

a. the probability of exactly 3 successes
b. the probability of 4 or more successes
c. the probability of exactly 7 successes
d. the expected value of the random variable

Please note nCx = n! / [(n-x)!*x!]

Binomial Probability = nCx * (p)x * (q)n-x, where n = number of trials and x is the number of successes.

Here n = 7, p =0.52, q = 1 – p = 0.48

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(a) P(X = 3) = 7C3 * (0.52)3 * (0.48)7-3 = 4 = 0.2612.

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(b) P(4 or More) = (X 4) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)

P(X = 4) = 7C4 * (0.52)4 * (0.48)7-4 = 3 = .0.2830

P(X = 5) = 7C5 * (0.52)5 * (0.48)7-5 = 2 = .0.1840

P(X = 6) = 7C6 * (0.52)6 * (0.48)7-6 = 1 = .0.0664

P(X = 7) = 7C7 * (0.52)7 * (0.48)7-7 = 0 = .0.0103

Therefore P(X > 4) = 0.2830 + 0.1840 + 0.0664 + 0.0103 = 0.5437

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(c) P(X = 7) = 7C7 * (0.52)7 * (0.48)7-7 = 0 = .0.0103

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(d) Expected value = n * p = 7 * 0.52 = 3.64

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