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Chapter 5: Problem 104

At the Bank of California, past data show that \(8 \%\) of all credit card holders default at some time in their lives. On one recent day, this bank issued 12 credit cards to new customers. Find the probability that of these 12 customers, eventually a. exactly 3 will default b. exactly 1 will default c. none will default

Short Answer

Expert verified
The probabilities that of the 12 customers, exactly 3 will default, exactly 1 will default and none will default are given by the calculations in steps 2, 3, and 4 respectively.

Step by step solution

01

Define Parameters

In this problem, the number of trials (\( n \)) is 12 (number of new customers), the probability of success (\( p \)) is 0.08 (probability a customer will default). Therefore, the formula for the binomial distribution becomes \( P(x) = C(12, x) * (0.08^x) * (0.92^(12 - x)) \).
02

Calculate Part - a

To find the probability that exactly 3 customers will default, substitute \( x = 3 \) into the binomial formula: \( P(3) = C(12, 3) * (0.08^3) * (0.92^9) \). Calculate the binomial coefficient \( C(12, 3) \) and the rest of the formula to find the probability.
03

Calculate Part - b

To find the probability that exactly 1 customer will default, substitute \( x = 1 \) into the binomial formula: \( P(1) = C(12, 1) * (0.08^1) * (0.92^11) \). Calculate the binomial coefficient \( C(12, 1) \) and the rest of the formula to find the probability.
04

Calculate Part - c

To find the probability that no customers will default, substitute \( x = 0 \) into the binomial formula: \( P(0) = C(12, 0) * (0.08^0) * (0.92^12) \). Calculate the binomial coefficient \( C(12, 0) \) and the rest of the formula to find the probability.

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Most popular questions from this chapter

Despite all efforts by the quality control department, the fabric made at Benton Corporation always contains a few defects. A certain type of fabric made at this corporation contains an average of \(.5\) defects per 500 yards. a. Using the Poisson formula, find the probability that a given piece of 500 yards of this fabric will contain exactly 1 defect. b. Using the Poisson probabilities table, find the probability that the number of defects in a given 500-yard piece of this fabric will be i. 2 to 4 ii. more than 3 iii. less than 2

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What are the conditions that must be satisfied to apply the Poisson probability distribution?
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