A biologist has 17 mice available for research. Unknown to her 7 are infected wi

Question

A biologist has 17 mice available for research. Unknown to her 7 are infected with a virus.   Suppose that she randomly selects 12 mice (without replacement) for her next project.  

(a) What is the probability density function for X = "the number of infected mice selected"?     i.e. what is P(X = x)?   

(b) Find the probability that exactly 3 of those selected are infected.

(c) Find the probability that more than 2 of those selected are infected.

(d) Find the mean (expected) number of infected mice in her sample. Also find the variance and standard deviation.

Explanation / Answer

N = 17

k = 7

n = 12

a) This is hypergeometric distribution

P(X = x) = kCx * N-kCn-x / NCn

or, P(X = x) = 7Cx * 10C12-x / 17C12

b) P(X = 3) =  7C3 * 10C9 / 17C12 = 0.0566

c) P(X > 2) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]

= 1 - [0 + 0 + 7C2 * 10C10 / 17C12 ]

= 1 - 0.0034

= 0.9964

d) mean = n * k / N = 12 * 7 / 17 = 4.94

Variance = n * k * (N - n) * (N - k) / (N2 * (N - 1)) = 12 * 7 * 5 * 10 / (17 * 17 * 16) = 0.9083

Stnadard deviation = sqrt(0.9083) = 0.9531

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