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) = 0.081 (round to three decimal places as needed.) (b) p(make your first sale on the first, second, or third call) = \boxed{} (round to three decimal places as needed.)

Explanation:

Step1: Recall Geometric Probability Formula

The probability of making a sale on the \(k\)-th call in a geometric distribution is \(P(X = k)=(1 - p)^{k - 1}p\), where \(p = 0.18\) (probability of success on a single call), and the probability of making a sale on the first, second, or third call is \(P(X\leq3)=P(X = 1)+P(X = 2)+P(X = 3)\).

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

For \(k = 1\), \(P(X = 1)=(1 - 0.18)^{1 - 1}\times0.18=(0.82)^{0}\times0.18 = 1\times0.18 = 0.18\).

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

For \(k = 2\), \(P(X = 2)=(1 - 0.18)^{2 - 1}\times0.18=(0.82)^{1}\times0.18 = 0.82\times0.18 = 0.1476\).

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

For \(k = 3\), \(P(X = 3)=(1 - 0.18)^{3 - 1}\times0.18=(0.82)^{2}\times0.18\). First, \((0.82)^{2}=0.6724\), then \(0.6724\times0.18 = 0.121032\).

Step5: Sum the Probabilities

\(P(X\leq3)=P(X = 1)+P(X = 2)+P(X = 3)=0.18 + 0.1476+0.121032 = 0.448632\). Round to three decimal places: \(0.449\).

Answer:

\(0.449\)