Question

the transaction history at an electronic goods store indicates that 21 percent of customers purchase the extended warranty when they buy an eligible item. suppose customers who buy eligible items are chosen at random, one at a time, until one is found who purchased the extended warranty. let the random variable x represent the number of customers it takes to find one who purchased the extended warranty. assume customers decisions on whether to purchase the extended warranty are independent. which of the following is closest to the probability that x > 3; that is, the probability that it takes more than 3 customers who buy an eligible item to find one who purchased the extended warranty?

a 0.131

b 0.390

c 0.493

d 0.507

e 0.624

Explanation:

Step1: Identify Distribution

X follows a geometric distribution (number of trials until first success), with success probability \( p = 0.21 \). The probability that \( X > k \) for a geometric distribution is \( (1 - p)^k \), since we need \( k \) failures before the first success.

Step2: Calculate \( P(X > 3) \)

For \( k = 3 \), this means the first 3 customers did not purchase the extended warranty. The probability of failure (not purchasing) is \( 1 - p = 1 - 0.21 = 0.79 \). So, \( P(X > 3) = (0.79)^3 \).

Step3: Compute the Value

Calculate \( 0.79^3 \):
\( 0.79 \times 0.79 = 0.6241 \)
\( 0.6241 \times 0.79 \approx 0.493 \) (Wait, no—wait, \( (0.79)^3 = 0.79 \times 0.79 \times 0.79 \). Wait, no, correction: Wait, \( X > 3 \) means the first 3 are failures, so \( (1 - p)^3 = 0.79^3 \). Let's recalculate:
\( 0.79^3 = 0.79 \times 0.79 = 0.6241 \); \( 0.6241 \times 0.79 = 0.6241 \times 0.79 \). Let's compute that:
\( 0.6241 \times 0.79 = 0.6241 \times (0.8 - 0.01) = 0.6241 \times 0.8 - 0.6241 \times 0.01 = 0.49928 - 0.006241 = 0.493039 \approx 0.493 \). Wait, but wait—no, wait, the geometric distribution: the probability that \( X > k \) is \( (1 - p)^k \), where \( k \) is the number of failures. Wait, \( X \) is the number of trials until first success, so \( X > 3 \) means the first 3 trials are failures (no purchase), so the 4th or later is the first success. So yes, \( P(X > 3) = (1 - 0.21)^3 = 0.79^3 \approx 0.493 \). Wait, but the options include 0.493 as option C. Wait, but wait, did I make a mistake? Wait, no—wait, let's check again. Wait, 0.79 cubed: 0.790.79=0.6241, 0.62410.79: 0.62410.7=0.43687, 0.62410.09=0.056169, sum is 0.43687 + 0.056169 = 0.493039, which is approximately 0.493. So the answer is C.

Answer:

C. 0.493