6. Johan has two bags. The first bag contains 2 red and 7 white balls, while the second contains 4 red and 5 white balls. One of the bags is selected randomly, and a ball is drawn from this bag. If the ball drawn is white, what is the probability that it is drawn from the second bag? A. $$ \frac{5}{12} $$ B. $$ \frac{1}{2} $$ C. $$ \frac{5}{9} $$ D. $$ \frac{7}{9} $$ E. $$ \frac{70}{81} $$

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UpStudy AI · Accepted Answer

To determine the probability that a white ball drawn is from the second bag, we'll use **Bayes' Theorem**.

**Given:**
- **Bag 1:** 2 red and 7 white balls (Total = 9 balls)
- **Bag 2:** 4 red and 5 white balls (Total = 9 balls)
- **Probability of choosing either bag:** $$ \frac{1}{2} $$

**Steps:**

1. **Calculate the probability of drawing a white ball from each bag:**
   - $$ P(\text{White} | \text{Bag 1}) = \frac{7}{9} $$
   - $$ P(\text{White} | \text{Bag 2}) = \frac{5}{9} $$

2. **Determine the total probability of drawing a white ball:**
   $$
   P(\text{White}) = P(\text{Bag 1}) \times P(\text{White} | \text{Bag 1}) + P(\text{Bag 2}) \times P(\text{White} | \text{Bag 2})
   $$
   $$
   P(\text{White}) = \frac{1}{2} \times \frac{7}{9} + \frac{1}{2} \times \frac{5}{9} = \frac{7 + 5}{18} = \frac{12}{18} = \frac{2}{3}
   $$

3. **Apply Bayes' Theorem to find the probability that the white ball is from Bag 2:**
   $$
   P(\text{Bag 2} | \text{White}) = \frac{P(\text{White} | \text{Bag 2}) \times P(\text{Bag 2})}{P(\text{White})}
   $$
   $$
   P(\text{Bag 2} | \text{White}) = \frac{\frac{5}{9} \times \frac{1}{2}}{\frac{2}{3}} = \frac{\frac{5}{18}}{\frac{12}{18}} = \frac{5}{12}
   $$

**Answer:**  
A. $$ \frac{5}{12} $$
The probability that the white ball is from the second bag is $$ \frac{5}{12} $$.

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