A wildlife biologist examines frogs for a genetic trait that

A wildlife biologist examines frogs for a genetic trait that may possibly be linked to industrial toxins in the environment. This trait is usually found in 1 of every 9 frogs, so the prevalence is 0.11. A sample of 11 frogs are collected and examined.

(a)The expected number of frogs in the sample with this trait is?

i tried 0.11*11, doesn\'t work, why?

(b)What is the probability that no more than 4 frogs in the sample have this trait?

(c)What is the probability that 3 or 4 frogs in the sample have this trait?

(d)What is the probability that at least 3 frogs in the sample have this trait?

Solution

a) E[X] = 0.11 * 11 = 1.21

since E[X] must be an integer hence expected number of frogs will be 1

b) P(success) = 0.11, P(failure) = 1 - 0.11 = 0.89

P( no more than 4 frogs in the sample have this trait) = P(0) + P(1) + P(2) + P(3)

= 11C0 * (0.89)^(11) + 11C1 * (0.89)^10 * (0.11) +11C2 * (0.89)^9 * (0.11)^2 + 11C3 * (0.89)^8 * (0.11)^3

= 0.9744

c) P(3 or 4 frogs) = P(3) + P(4)

= 11C4 * (0.89)^7 * (0.11)^4 + 11C3 * (0.89)^8 * (0.11)^3

d) P(at least 3 frogs) = 1 - P(0) - P(1) - P(2)

= 1 - 11C0 * (0.89)^(11) + 11C1 * (0.89)^10 * (0.11) +11C2 * (0.89)^9 * (0.11)^2

= 1 -0.887

= 0.113