Problem 27 - Basic Statistics for Business and Economics with Student

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 3 loans will be defaulted? b. What is the probability that at least 3 loans will be defaulted?

Step-by-Step Solution

Answer

a. \( P(X = 3) \approx 0.0563 \); b. \( P(X \geq 3) \approx 0.0702 \).

Step 1: Define the Probability Parameters

First, identify the parameters of the problem related to probability. The probability that a single loan will be defaulted is \( p = 0.025 \). The number of loans (trials) is \( n = 40 \). We need to find the probabilities of defaulting for 3 loans.

Step 2: Choose the Appropriate Probability Distribution

Since each loan can be treated as an independent event where it is either defaulted or not, we use the binomial distribution to find the probabilities. The formula for the binomial probability is: \( P(X = k) = \binom{n}{k}p^k(1-p)^{n-k} \) where \( X \) is the random variable representing the number of successful defaults.

Step 3: Calculate the Probability of 3 Loans Being Defaulted

Using the binomial formula with \( n = 40 \), \( k = 3 \), and \( p = 0.025 \),\[P(X = 3) = \binom{40}{3}(0.025)^3(1-0.025)^{40-3}\]Calculate the binomial coefficient \( \binom{40}{3} \) and substitute the values into the formula to find \( P(X = 3) \).

Step 4: Calculate the Probability of At Least 3 Loans Being Defaulted

The probability of at least 3 defaults is the cumulative probability of 3 or more defaults. This can be expressed as \( P(X \geq 3) = 1 - P(X < 3) \). First, calculate \( P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \) using the binomial distribution formula for each outcome.

Step 5: Perform Final Calculation for Part (b)

Using the probabilities from Step 4, calculate: \[P(X \geq 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]\]Substitute the computed values to find the probability that at least 3 loans will be defaulted.

Key Concepts

Binomial Distribution

The binomial distribution is a fundamental probability distribution that models the number of successes in a fixed number of independent trials of a binary experiment. Imagine flipping a coin many times; each flip can result in heads (success) or tails (failure).
For modeling scenarios like the loan defaults in the exercise, where each loan independently can either default or not, the binomial distribution becomes highly relevant. Here, key elements of the binomial distribution are used:

• The number of trials (\( n \)) which is the number of loans (40).
• The probability of success (\( p \)), in this case, a single loan defaulting, which is 0.025.
• The number of successful outcomes (\( k \)) you are interested in finding the probability for, like 3 defaults.

Using the binomial probability formula, you can calculate how likely it is that a certain number of defaults occur within those loans. The formula is given by\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}.\]This allows you to calculate the specific probabilities related to the problem, like finding the chance exactly 3 loans out of 40 will go unpaid.

Cumulative Probability

Cumulative probability assesses the probability that a random variable is less than or equal to a certain value. It's incredibly useful when answering questions that start with "at least." In our problem, the cumulative probability helps in finding the chance that at least 3 loans will default.
To calculate this, first find the probability of fewer than 3 defaults occurring (0, 1, or 2 defaults). Calculate these probabilities individually using the binomial distribution formula:

• \( P(X = 0) \)
• \( P(X = 1) \)
• \( P(X = 2) \)

Once you have these probabilities, the cumulative probability of having at least 3 defaults is found by subtracting the sum of these probabilities from 1:\[P(X \geq 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)].\]This method gives you the probability of at least 3 loans not being repaid, considering all possibilities of three or more defaults.

Independent Events

Independence is a key concept in probability where one event does not affect the outcome of another. In the context of our loan example, it means the outcome of one loan (whether it defaults or not) does not influence another.
This assumption allows the use of the binomial distribution, where each trial (loan) has only two possible outcomes and is unaffected by any other loans. Each loan has an independent probability of defaulting, ensuring calculations are straightforward using the binomial formula.
For real-world problems, assuming independence simplifies complex systems and allows mathematicians and statisticians to model random occurrences effectively. Although perfect independence might not always hold in reality, it is a good approximation in many scenarios including small default probabilities across a large number of loans.