Find the indicated probabilities using the geometric distrib

Find the indicated probabilities using the geometric distribution or Poisson distribution. Then determine if the events are unusual. If convenient, use a Poisson probability table or technology to find the probabilities.

Assume the probability that you will make a sale on any given telephone call is 0.21. Find the probability that you (a) make your first sale on the fifth call, (b) make your sale on the first, second, or third, and (c) do not make a sale on the first three calls.

Solution

Let X denote the number of calls required to make the first sale.

Then X~ Geometric(p = 0.21) where p is the probability that a sale is made on any telephone call.

Then the pmf of X is

f(x) = (1-p)^x-1p if x=1,2,3,.....

= 0 otherwise

(a)

P[X=5]

= (1-p)⁴p

= 0.79⁴ * 0.21

=0.0818

(b)

P[X=1,2,3]

=P[X=1]+P[X=2]+P[X=3]

= p + (1-p)p + (1-p)²p

= 0.21 + 0.79*0.21 + 0.79²*0.21

=0.5069

(c)

P[X> 3]

=1 - P[X=1,2,3]

= 1-0.5069

= 0.4931