Question

alexandria, the owner of the speedy stop convenience store is trying to decide whether to discontinue selling magazines. alexandria suspects that only 8% of her customers buy a magazine and she thinks that she might be able to use the space to sell something more profitable.
before making a final decision, she decides that for one day she will keep track of the number of customers that buy a magazine. assuming alexandrias suspicion that 8% of her customers buy a magazine is correct, what is the probability that out of the first 17 customers buy a magazine? (give result accurate to at least 3 decimal places.)

Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success in a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\). Here, \(n = 17\), \(p=0.08\), and we want to find \(P(X\geq1)=1 - P(X = 0)\)

Step2: Calculate \(P(X = 0)\)

For \(k = 0\), \(C(17,0)=\frac{17!}{0!(17-0)!}=1\), \(p = 0.08\), \(1-p=0.92\). Then \(P(X = 0)=C(17,0)\times(0.08)^{0}\times(0.92)^{17}\)

$$P(X = 0)=1\times1\times0.92^{17}$$

Using a calculator, \(0.92^{17}\approx0.249\)

Step3: Calculate \(P(X\geq1)\)

\(P(X\geq1)=1 - P(X = 0)=1-0.249 = 0.751\)

Answer:

\(0.751\)