Question

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

Answer 1

Please note ⁿC_x = n! / [(n-x)!*x!]

Binomial Probability = ⁿC_x * (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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