A box contains 4 white and 5 black balls. A ball is drawn at random and its colour is noted. A ball is then put back in the box along with two additional balls of its opposite colour. If a ball is drawn again from the box, then the probability that the ball drawn now is black, is

Options

A. `(7)/(11)`
B. `(5)/(11)`
C. `(53)/(99)`
D. `(48)/(99)`

Explanation

To solve the problem step by step, we will calculate the probability that the second ball drawn is black after following the described process.

### Step 1: Determine the probabilities of drawing each color on the first draw.
- The box contains 4 white balls and 5 black balls, making a total of 9 balls.
- The probability of drawing a black ball (E1) on the first draw is:
\[ P(E1) = \frac{\text{Number of black balls}}{\text{Total number of balls}} = \frac{5}{9} \]
- The probability of drawing a white ball (E2) on the first draw is:
\[ P(E2) = \frac{\text{Number of white balls}}{\text{Total number of balls}} = \frac{4}{9} \]

### Step 2: Analyze the outcomes based on the first draw.
- **If a black ball is drawn (E1):**
- We put back the black ball and add 2 white balls.
- The new composition of the box will be 5 black balls and 6 white balls.
- The probability of drawing a black ball on the second draw now is:
\[ P(\text{Black on 2nd draw | E1}) = \frac{5}{11} \]

- **If a white ball is drawn (E2):**
- We put back the white ball and add 2 black balls.
- The new composition of the box will be 7 black balls and 4 white balls.
- The probability of drawing a black ball on the second draw now is:
\[ P(\text{Black on 2nd draw | E2}) = \frac{7}{11} \]

### Step 3: Use the law of total probability to find the overall probability of drawing a black ball on the second draw.
- The total probability of drawing a black ball on the second draw can be calculated as:
\[ P(\text{Black on 2nd draw}) = P(E1) \cdot P(\text{Black on 2nd draw | E1}) + P(E2) \cdot P(\text{Black on 2nd draw | E2}) \]
Substituting the values:
\[ P(\text{Black on 2nd draw}) = \left(\frac{5}{9} \cdot \frac{5}{11}\right) + \left(\frac{4}{9} \cdot \frac{7}{11}\right) \]

### Step 4: Calculate the total probability.
- Calculate each term:
\[ P(\text{Black on 2nd draw}) = \frac{25}{99} + \frac{28}{99} = \frac{53}{99} \]

### Final Answer:
Thus, the probability that the ball drawn now is black is:
\[ \frac{53}{99} \]

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