Answer to Question #2826 in Statistics and Probability for alyssa

Question

20% of the deer in a wildlife preservation part are believed to have certain antler infections. A zoologist randomly captures 15 deer and checks their antlers. Assuming the prediction is accurate, determine the probability that more than three of the deer will have infected antlers.

Answer 1

The number X of deers among n=15 ones having infected antlers has binomial distribution with parameters
p=0.2, q=1-p=0.8,
so
P(X=m) = C(15, m) 0.2^m 0.8^(15-m)
Hence
P(X>3) = 1-P(X<4) = 1- P(X=0) - P(X=1) - P(X=2) - P(X=3) =
= 1- 0.8¹⁵ - 15∙0.2∙0.8¹⁴- (15∙14)/2∙0.2²∙0.8¹³ - (15∙14∙13)/(2∙3)∙0.2³∙0.8¹² =
= 1 - 0.03518 - 0.13194 - 0.23090 - 0.25014 = 0.35184