One out of every 5000 individuals in a population carries a certain defective gene. A random sample of 1000 individuals is studied.
a. What is the probability that exactly one of the sample individuals carries the gene?
b. What is the probability that none of the sample individuals carries the gene?
c. What is the probability that more than two of the sample individuals carry the gene?
d. What is the mean of the number of sample individuals that carry the gene?
e. What is the standard deviation of the number of sample individuals that carry the gene?
Let X be the number of individuals in the sample that carry the gene. Then X is the number of successes in n= 1000 Bernoulli trials, each of which has success probability p = 0.0002. The mean of X is np= (1000)(0.0002) = 0.2. Since n is large and p is small, X ~ Poisson(0.2) to a very close approximation.
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