Question

a survey found that customers are overcharged by price scanning systems, on average, on 2.25% of items. suppose a customer purchases 12 items. find the following probability.
a customer is overcharged on 1 item.
the probability is
(round to six decimal places as needed)

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 on a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Determine the values of \(n\), \(k\), and \(p\)

Here, \(n = 12\) (number of items), \(k = 1\) (number of over - charged items), and \(p=0.0225\) (probability of an item being over - charged). Then \(1-p = 1-0.0225=0.9775\)

Step3: Calculate the combination \(C(n,k)\)

$$
LATEXBLOCK0
$$

Step4: Calculate the probability \(P(X = 1)\)

$$
LATEXBLOCK1
$$

First, calculate \((0.9775)^{11}\approx0.777797\)
Then \(P(X = 1)=12\times0.0225\times0.777797 = 12\times0.0175004325=0.210005\)

Answer:

\(0.210005\)