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A local bank reviewed its credit card policy with the intention of recalling some of its credit cards. In the past approximately \(5 \%\) of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of .05 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is .20 for customers who do not default. Of course, the probability of missing a monthly payment for those who default is 1 a. Given that a customer missed one or more monthly payments, compute the posterior probability that the customer will default. b. The bank would like to recall its card if the probability that a customer will default is greater than \(.20 .\) Should the bank recall its card if the customer misses a monthly payment? Why or why not?

Step by step solution

01

Define the Problem

We want to find the posterior probability that a customer will default given that they missed a payment. This is essentially finding \( P(D|M) \): the probability of default \( D \) given a missed payment \( M \). This requires using Bayes' theorem.

02

Use Bayes' Theorem

Bayes' theorem is represented as \[ P(D|M) = \frac{P(M|D) \cdot P(D)}{P(M)} \]. We know that \( P(D) = 0.05 \), \( P(M|D) = 1 \) since if they default, they will definitely miss a payment. The only unknown is \( P(M) \), the overall probability a customer misses a payment.

03

Calculate P(M)

Use the law of total probability: \( P(M) = P(M|D) \cdot P(D) + P(M|eg D) \cdot P(eg D) \). With \( P(eg D) = 1 - P(D) = 0.95 \) and \( P(M|eg D) = 0.20 \), we find \[ P(M) = 1 \cdot 0.05 + 0.20 \cdot 0.95 = 0.24 \].

04

Plug Values into Bayes' Theorem

Now substitute the known values into Bayes' theorem: \[ P(D|M) = \frac{1 \cdot 0.05}{0.24} = \frac{0.05}{0.24} = 0.2083 \].

05

Decision Based on Posterior Probability

The bank wants to recall cards if the probability of default is higher than 0.2. Since 0.2083 > 0.2, this probability is slightly above 0.2, suggesting that a recall is justified based solely on this probability.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bayes' Theorem

Bayes' Theorem is a powerful mathematical formula used for calculating conditional probabilities. This theorem enables us to find the probability of an event, based on prior knowledge of conditions that might be related to the event. In the context of the bank exercise, we want to decide whether to recall a credit card by finding out how likely it is for a customer to default if they miss a payment. To do this, we use Bayes' Theorem, which is represented as follows:

• \[ P(D|M) = \frac{P(M|D) \cdot P(D)}{P(M)} \]

Where:

• \( P(D|M) \) is the posterior probability that a customer will default given that they missed a payment.
• \( P(M|D) \) is the probability of missing a payment given that the customer defaults.
• \( P(D) \) is the initial probability that a customer will default (prior probability).
• \( P(M) \) is the total probability that a payment is missed.

Using Bayes' Theorem allows us to take concrete steps in decision-making based on updated beliefs given new evidence. This is especially useful in fields like finance, medicine, and machine learning.

Posterior Probability

Posterior Probability is a core concept in Bayesian inference and provides the updated probability of a hypothesis after considering new evidence. This new probability takes into account the prior probability and the likelihood of the observed data.
In the bank's situation, the posterior probability, \( P(D|M) \), is essential to gauge the likelihood of default once a payment is missed. Initially, the bank had a prior default probability, \( P(D) = 0.05 \); however, after observing a missed payment event, this probability gets updated.
To compute this, we need:

• The prior probability, \( P(D) \), representing former beliefs about default likelihood.
• The likelihood of observing a missed payment given default, \( P(M|D) \), which is given as 1, indicating certainty.
• The total probability of missing a payment, \( P(M) \), is calculated using all possibilities that lead to a missed payment.

The resulting posterior probability is a more informed basis for recalling the card, closely approximating or exceeding the threshold set by the bank, thus guiding actionable decisions.

Law of Total Probability

The Law of Total Probability is a fundamental principle that aids in determining the overall probability of an event by considering all possible ways that event can happen. It is particularly useful when calculating the probability of compound events.
In our credit card scenario, the law helps calculate \( P(M) \), which is the probability of missing a payment irrespective of default status. This total probability is computed by:

• Identifying the probability of missing a payment due to default, \( P(M|D) \cdot P(D) \).
• Adding the probability of missing a payment when not defaulting, \( P(M| eg D) \cdot P( eg D)\).

In simplified form:

• \( P(M) = P(M|D) \cdot P(D) + P(M| eg D) \cdot P( eg D) \)

For our case,

• \( P(M|D) = 1 \), \( P(D) = 0.05 \), and \( P(M| eg D) = 0.20 \).
• \( eg D \) is the probability of not defaulting, hence \( 1 - 0.05 \).

Adding these possibilities provides \( P(M) = 0.24 \). This helps inform the bank's decision-making process when applying Bayes' theorem to calculate \( P(D|M) \).