Question

A flycatcher is trying to catch passing bugs. The probability that it catches a bug on any given try is 20% a. What is the probability that it catches its first 7. bug on its fourth try? b. What is the probability that it catches its first bug after at least four failures, assuming that it keeps trying until it is successful?

Answer 1

a) Let X denotes the number of try to catch its first bug.

X ~ Geometric(0.20)

The probability mass function of X is

$P(X=x) = 0.2*(1-0.2)^{x-1},x=1,2,3,...$

Now,

$P(X=4) = 0.2*(1-0.2)^{4-1} = 0.5904$

b) Let Y denotes the number of failures to catch its first bug.

The probability mass function of X is

$P(Y=y) = 0.2*(1-0.2)^{y},y=0,1,2,3,...$

Now,

$P(X\geq 4) = 1 - P(X<4) = 1-P(X\leq 3) =1 - 0.5904 = 0.4096$