Questions: Suppose in a certain year, the probability that a person in the United States would declare personal bankruptcy was 0.0017. The probability that a person in the United States would declare personal bankruptcy and had recently experienced medical-related issues was 0.0013. What was the probability that a person had recently experienced medical-related issues, given that they had declared personal bankruptcy? (Round your answer to two decimal places.)

Suppose in a certain year, the probability that a person in the United States would declare personal bankruptcy was 0.0017. The probability that a person in the United States would declare personal bankruptcy and had recently experienced medical-related issues was 0.0013. What was the probability that a person had recently experienced medical-related issues, given that they had declared personal bankruptcy? (Round your answer to two decimal places.)

Transcript text: Suppose in a certain year, the probability that a person in the United States would declare personal bankruptcy was 0.0017. The probability that a person in the United States would declare personal bankruptcy and had recently experienced medical-related issues was 0.0013. What was the probability that a person had recently experienced medical-related issues, given that they had declared personal bankruptcy? (Round your answer to two decimal places.)

Solution

Solution Steps

To find the probability that a person had recently experienced medical-related issues given that they had declared personal bankruptcy, we can use the concept of conditional probability. The formula for conditional probability is:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Where:

\( P(A|B) \) is the probability of event A occurring given that event B has occurred.
\( P(A \cap B) \) is the probability of both events A and B occurring.
\( P(B) \) is the probability of event B occurring.

In this case:

Event A is "recently experienced medical-related issues."
Event B is "declared personal bankruptcy."

Given:

\( P(A \cap B) = 0.0013 \)
\( P(B) = 0.0017 \)

We need to calculate \( P(A|B) \).

Solution Approach

Use the formula for conditional probability.
Substitute the given values into the formula.
Perform the division to find the result.
Round the result to two decimal places.

Step 1: Identify Given Probabilities

We are given the following probabilities:

The probability that a person in the United States would declare personal bankruptcy, \( P(B) = 0.0017 \).
The probability that a person in the United States would declare personal bankruptcy and had recently experienced medical-related issues, \( P(A \cap B) = 0.0013 \).

Step 2: Apply the Conditional Probability Formula

To find the probability that a person had recently experienced medical-related issues given that they had declared personal bankruptcy, we use the conditional probability formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Step 3: Substitute the Given Values

Substitute the given values into the formula: \[ P(A|B) = \frac{0.0013}{0.0017} \]

Step 4: Perform the Division

Perform the division to find the result: \[ P(A|B) = 0.7647 \]

Step 5: Round the Result

Round the result to two decimal places: \[ P(A|B) \approx 0.76 \]