Question

find the indicated probabilities using the geometric distribution, the poisson distribution, or the binomial distribution. then determine if the events are unusual. if convenient, use the appropriate probability table or technology to find the probabilities.
assume the probability that you will make a sale on any given telephone call is 0.18. find the probability that you (a) make your first sale on the fifth call, (b) make your first sale on the first, second, or third call, and (c) do not make a sale on the first three calls.
(a) p(make your first sale on the fifth call) =
(round to three decimal places as needed.)

Explanation:

Step1: Recall Geometric Probability Formula

The geometric distribution formula for the probability of the first success (sale) on the \( n \)-th trial is \( P(X = n)=(1 - p)^{n - 1}p \), where \( p \) is the probability of success on a single trial. Here, \( p = 0.18 \) and \( n = 5 \).

Step2: Calculate \( (1 - p)^{n - 1} \)

First, find \( 1 - p \): \( 1 - 0.18 = 0.82 \). Then, \( n - 1 = 5 - 1 = 4 \). So, \( (0.82)^{4} \). Calculate \( 0.82^4 \): \( 0.82\times0.82 = 0.6724 \), \( 0.6724\times0.82 = 0.551368 \), \( 0.551368\times0.82 \approx 0.45212176 \).

Step3: Multiply by \( p \)

Multiply the result from Step 2 by \( p = 0.18 \): \( 0.45212176\times0.18 \approx 0.0813819168 \).

Step4: Round to Three Decimal Places

Rounding \( 0.0813819168 \) to three decimal places gives \( 0.081 \).

Answer:

\( 0.081 \)