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 0.11. 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) = 0.069
(round to three decimal places as needed.)
(b) p(make your first sale on the first, second, or third call) =
(round to three decimal places as needed.)

Explanation:

Step1: Recall geometric distribution formula

The probability mass function of a geometric distribution is \(P(X = k)=(1 - p)^{k - 1}p\), where \(p\) is the probability of success on a single trial and \(k\) is the number of trials until the first success.

Step2: Calculate \(P(X = 1)\)

For \(k = 1\), \(P(X=1)=(1 - 0.11)^{1 - 1}\times0.11=0.11\)

Step3: Calculate \(P(X = 2)\)

For \(k = 2\), \(P(X = 2)=(1 - 0.11)^{2 - 1}\times0.11=(0.89)\times0.11 = 0.0979\)

Step4: Calculate \(P(X = 3)\)

For \(k = 3\), \(P(X = 3)=(1 - 0.11)^{3 - 1}\times0.11=(0.89)^{2}\times0.11=0.7921\times0.11 = 0.087131\)

Step5: Use addition rule for mutually - exclusive events

Since the events \(X = 1\), \(X = 2\), and \(X = 3\) are mutually - exclusive, \(P(X\leq3)=P(X = 1)+P(X = 2)+P(X = 3)\)
\(P(X\leq3)=0.11+0.0979 + 0.087131=0.295031\approx0.295\)

Answer:

\(0.295\)