A bag contains 4 white and 5 black balls. Another bag contains 9 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.

Options

A. `9/17`
B. `5/9`
C. `10/17`
D. `7/9`

Explanation

To solve the problem step by step, we will calculate the probability of drawing a white ball from the second bag after transferring one ball from the first bag.

### Step 1: Understand the contents of the bags
- **Bag 1**: Contains 4 white balls and 5 black balls (Total = 9 balls).
- **Bag 2**: Contains 9 white balls and 7 black balls (Total = 16 balls).

### Step 2: Calculate the probability of transferring a white ball from Bag 1 to Bag 2
The probability of transferring a white ball from Bag 1 is given by:
\[ P(\text{White from Bag 1}) = \frac{\text{Number of white balls in Bag 1}}{\text{Total balls in Bag 1}} = \frac{4}{9} \]

### Step 3: Calculate the probability of drawing a white ball from Bag 2 after transferring a white ball
If a white ball is transferred, Bag 2 will then contain:
- White balls = 9 + 1 = 10
- Black balls = 7
- Total = 10 + 7 = 17

The probability of drawing a white ball from Bag 2 is:
\[ P(\text{White from Bag 2 | White transferred}) = \frac{10}{17} \]

### Step 4: Calculate the probability of transferring a black ball from Bag 1 to Bag 2
The probability of transferring a black ball from Bag 1 is given by:
\[ P(\text{Black from Bag 1}) = \frac{\text{Number of black balls in Bag 1}}{\text{Total balls in Bag 1}} = \frac{5}{9} \]

### Step 5: Calculate the probability of drawing a white ball from Bag 2 after transferring a black ball
If a black ball is transferred, Bag 2 will then contain:
- White balls = 9
- Black balls = 7 + 1 = 8
- Total = 9 + 8 = 17

The probability of drawing a white ball from Bag 2 is:
\[ P(\text{White from Bag 2 | Black transferred}) = \frac{9}{17} \]

### Step 6: Use the law of total probability to find the overall probability of drawing a white ball from Bag 2
The total probability of drawing a white ball from Bag 2 is given by:
\[ P(\text{White from Bag 2}) = P(\text{White from Bag 1}) \cdot P(\text{White from Bag 2 | White transferred}) + P(\text{Black from Bag 1}) \cdot P(\text{White from Bag 2 | Black transferred}) \]
Substituting the values:
\[ P(\text{White from Bag 2}) = \left(\frac{4}{9} \cdot \frac{10}{17}\right) + \left(\frac{5}{9} \cdot \frac{9}{17}\right) \]

### Step 7: Calculate the two parts of the equation
Calculating the first part:
\[ \frac{4}{9} \cdot \frac{10}{17} = \frac{40}{153} \]
Calculating the second part:
\[ \frac{5}{9} \cdot \frac{9}{17} = \frac{45}{153} \]

### Step 8: Add the two probabilities together
\[ P(\text{White from Bag 2}) = \frac{40}{153} + \frac{45}{153} = \frac{85}{153} \]

### Final Answer
Thus, the probability that the ball drawn from the second bag is white is:
\[ \frac{85}{153} \]