Question

Twenty-five percent of the customers entering a grocery store between 5 P.M. and 7 P.M. use an express checkout. Consider five randomly selected customers, and let \(x\) denote the number among the five who use the express checkout. a. What is \(p(2)\), that is, \(P(x=2)\) ? b. What is \(P(x \leq 1)\) ? c. What is \(P(2 \leq x)\) ? (Hint: Make use of your computation in Part (b).) d. What is \(P(x \neq 2) ?\)

Short answer

The answer involves four parts as it's a multi-part problem: Part a, compute the binomial probability with \(x=2\); Part b, sum the probabilities for \(x=0\) and \(x=1\); Part c, subtract the result from Part b from 1; And for Part d, subtract computation from part a from 1.

Step-by-step solution

Calculation for Part a

To calculate \(P(x=2)\), we'll use the binomial distribution probability formula, with \(n=5\), \(x=2\), and \(p=0.25\). The formula becomes \(P(2; 5, 0.25) = C(5, 2) * 0.25^2 * (1 - 0.25)^{5 - 2}\). Doing the calculation gives the answer for Part a.

Calculation for Part b

Part b asks for \(P(x \leq 1)\), which means the probability that either 0 or 1 customer uses the express checkout. In this case, we can calculate the sum of the binomial probabilities for \(x=0\) and \(x=1\) over the 5 customers. This requires two sub-calculations: \(P(0; 5, 0.25) = C(5, 0) * 0.25^0 * (1 - 0.25)^{5 - 0}\) and \(P(1; 5, 0.25) = C(5, 1) * 0.25^1 * (1 - 0.25)^{5 - 1}\). Add these two probabilities to get the answer for Part b.

Calculation for Part c

This requires calculation for \(P(2 \leq x)\), that is the probability that at least 2 customers use the express checkout. This computation can be reduced by 1-the probability computed in Part b (because the sum of all probabilities equals 1). So, \(P(2 \leq x)= 1 -P(x \leq 1)\). This opposite event method helps us avoid calculating each individual probability for \(x = 2,3,4,5\).

Calculation for Part d

To solve for \(P(x \neq 2)\), we can use the concept of complementary probability again. This event is the opposite of \(x = 2\), so we can compute this probability by \(P(x \neq 2) = 1 - P(x = 2)\), where \(P(x = 2)\) was calculated in part a.