Question

the probability a car salesman sells a car to a customer is 0.05
assuming the salesmen sees 12 customers in a week, what is the probability he sells less than 2 cars?
write answer using three decimal places

Explanation:

Step1: Identify the distribution

This is a binomial distribution problem. 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, \(n = 12\), \(p=0.05\), and we want \(P(X\lt2)=P(X = 0)+P(X = 1)\)

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

For \(k = 0\):

$$
LATEXBLOCK0
$$

Using a calculator, \((0.95)^{12}\approx0.54036\)

Step3: Calculate \(P(X = 1)\)

For \(k = 1\):

$$
LATEXBLOCK1
$$

\((0.95)^{11}\approx0.5688\), so \(12\times0.05\times0.5688 = 0.34128\)

Step4: Calculate \(P(X\lt2)\)

\(P(X\lt2)=P(X = 0)+P(X = 1)=0.54036+0.34128 = 0.88164\approx0.882\)

Answer:

\(0.882\)