Question

video lesson 5.2 example 7 – finding a conditional probability

1. at a clothing store, 75% of the customers buy pants. only 20% of the customers buy pants and a belt.

what is the probability that a customer who buys pants also buys a belt?

Explanation:

Step1: Recall Conditional Probability Formula

The formula for conditional probability is \( P(B|A) = \frac{P(A \cap B)}{P(A)} \), where \( P(B|A) \) is the probability of event \( B \) given event \( A \), \( P(A \cap B) \) is the probability of both \( A \) and \( B \) occurring, and \( P(A) \) is the probability of event \( A \) occurring.
Here, let \( A \) be the event that a customer buys pants, and \( B \) be the event that a customer buys a belt. So we need to find \( P(B|A) \).

Step2: Identify Given Probabilities

We know that \( P(A) = 0.75 \) (75% of customers buy pants) and \( P(A \cap B) = 0.20 \) (20% of customers buy pants and a belt).

Step3: Apply the Formula

Substitute the values into the conditional probability formula: \( P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.20}{0.75} \).

Step4: Simplify the Fraction

Simplify \( \frac{0.20}{0.75} \). Multiply numerator and denominator by 100 to eliminate decimals: \( \frac{20}{75} \). Then simplify by dividing numerator and denominator by 5: \( \frac{4}{15} \approx 0.2667 \) (or 26.67%).

Answer:

The probability is \(\frac{4}{15}\) (or approximately 26.67%).