Problem 27 - Basic Statistics for Business & Economics

Question

Ms. Bergen is a loan officer at Coast Bank and Trust. From her years of experience, she estimates that the probability is .025 that an applicant will not be able to repay his or her installment loan. Last month she made 40 loans. a. What is the probability that three loans will be defaulted? b. What is the probability that at least three loans will be defaulted?

Step-by-Step Solution

Answer

a) The probability of exactly three defaults is approximately 0.212. b) The probability of at least three defaults is approximately 0.263.

Step 1: Understanding the Problem

We need to find out the probability that exactly three out of the 40 loans default, and then the probability that at least three loans default. This is a binomial probability situation because there are two possible outcomes for each loan – it either defaults or it does not default.

Step 1: Define the Parameters

Identify the parameters of the binomial distribution: - The number of trials (n), which is the number of loans, is 40. - The probability of success (p), where success means a loan defaults, is 0.025. - The number of successes (k), or number of defaults we are looking for, is 3 in part (a).

Step 2: Binomial Formula for Exactly k Successes

The probability of getting exactly k successes in a binomial distribution is given by the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Substitute the known values: \[ P(X = 3) = \binom{40}{3} (0.025)^3 (0.975)^{37} \]

Step 3: Calculate Binomial Coefficient

Calculate the binomial coefficient \( \binom{40}{3} \): \[ \binom{40}{3} = \frac{40!}{3!(40-3)!} = \frac{40 \times 39 \times 38}{3 \times 2 \times 1} = 9880 \]

Step 4: Calculate Probability of Exactly Three Defaults

Substitute \( \binom{40}{3} \) back into the probability formula:\[ P(X = 3) = 9880 \times (0.025)^3 \times (0.975)^{37} \]Perform the calculations to find the probability.

Step 5: Calculate Probability from Cumulative Distribution

To find the probability of at least three defaults, calculate:\[ P(X \geq 3) = 1 - P(X < 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2)) \]Calculate each using the binomial formula and then subtract from 1.

Step 6: Probability of Fewer Than Three Defaults

Calculate each term \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \) similarly to step 4, using \( k = 0, 1, \text{ and } 2 \), then sum those probabilities.

Step 7: Final Calculation

Subtract the sum from step 6 from 1 to find the probability of at least three defaults:\[ P(X \geq 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2)) \]This completes the calculation for part (b).

Key Concepts

Probability Theory in Context

Probability theory helps us predict the likelihood of different outcomes. In this problem, probability is used to estimate the chance of loans defaulting.
Ms. Bergen, a loan officer, uses her experience to predict that each loan will default with a probability of 0.025. This is a basic application of probability theory.
The principles allow us to make informed guesses about future events, based on patterns or historical data. When dealing with a series of trials (like loans), we use these probabilities to explore possible numbers of defaults, which is made easier by the binomial distribution model.
The model relies on:

• The number of trials (n), which are the loans given out, totalling 40.
• A consistent outcome probability (p), here 0.025 for each loan defaulting.
• Each trial being independent, meaning the outcome of one loan doesn't affect another.

These elements of probability theory make it possible to work out the chances of exactly three defaults, or more, as asked in the problem.

Understanding the Binomial Coefficient

The binomial coefficient is an important part of binomial distribution calculations. It represents the number of ways to choose a certain number of successes (like loan defaults) from a total number of trials (such as number of loans).
The coefficient is written as \( \binom{n}{k} \), where \( n \) is the total number of trials, and \( k \) is the number of successful outcomes. In this problem, Ms. Bergen is looking to calculate when 3 loans default out of 40.
With the formula \( \binom{40}{3} = \frac{40!}{3!(40-3)!} \), you find that there are 9880 different combinations where exactly 3 loans can default. The factorial expression \( n! \) (which is the product of all numbers up to \( n \)) helps compute this number.
Understanding how to calculate the binomial coefficient allows us to distribute probabilities over possible outcomes accurately. It makes sure we consider all possible combinations that can occur when calculating probabilities.

Exploring the Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) is used to find the probability of a variable taking on a value less than or equal to a specific value.
In the context of the binomial distribution, the CDF gives us the probability of up to a certain number of defaulting loans happening. In Ms. Bergen's problem, we use it to find the probability of at least three loans defaulting.
By calculating \( P(X \geq 3) \), we consider all cases where 3 or more loans default. However, computing this directly for each outcome would be extensive, so we subtract the CDF up to 2 from 1 (i.e., \( 1 - (P(X = 0) + P(X = 1) + P(X = 2)) \)).
This method efficiently leverages the properties of the CDF, simplifying the calculation of probabilities for consecutive outcomes and turning a labour-intensive process into manageable steps.
The CDF adjusts for cumulative probabilities, aiding in understanding the distribution of outcomes over probability space. This is essential for interpreting broader outcomes, beyond exact matches.