A bag contains 3 red and 4 black balls and another bag has 4 red and 2 black balls . One bag is selected at random and from the selected at random and from the selected bag a ball is drawn .Let E be the event the first bag is selected F be the event that the second bag is selected ,G be the event that ball drawn is red .
find `P(G/F)`

Options

A. `2/3`
B. `1/9`
C. `5/9`
D. `4/5`

Answer

To solve the problem step by step, we need to find \( P(G | F) \), which is the probability of drawing a red ball given that the second bag is selected.

### Step 1: Understand the Events
- Let \( E \) be the event that the first bag is selected.
- Let \( F \) be the event that the second bag is selected.
- Let \( G \) be the event that a red ball is drawn.

### Step 2: Determine the Total Number of Balls in Each Bag
- **First Bag**: Contains 3 red and 4 black balls. Total = \( 3 + 4 = 7 \) balls.
- **Second Bag**: Contains 4 red and 2 black balls. Total = \( 4 + 2 = 6 \) balls.

### Step 3: Find the Probability of Selecting the Second Bag
Since one of the two bags is selected at random:
\[ P(F) = \frac{1}{2} \]

### Step 4: Find the Probability of Drawing a Red Ball from the Second Bag
If the second bag is selected, the probability of drawing a red ball from it is:
\[ P(G | F) = \frac{\text{Number of red balls in the second bag}}{\text{Total number of balls in the second bag}} = \frac{4}{6} = \frac{2}{3} \]

### Step 5: Conclusion
Thus, the probability of drawing a red ball given that the second bag is selected is:
\[ P(G | F) = \frac{2}{3} \]

### Final Answer
The final answer is \( \frac{2}{3} \).

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