A bag contains 12 white pearls and 18 black pearls. Two pearls are drawn in succession without replacement. The probability that the first pearl is white and the second is black, is

Options

A. `32/145`
B. `28/143`
C. `36/145`
D. `36/143`

Answer

To solve the problem, we need to find the probability that the first pearl drawn is white and the second pearl drawn is black, without replacement.

### Step-by-Step Solution:

1. **Identify the Total Number of Pearls:**
- The bag contains 12 white pearls and 18 black pearls.
- Total pearls = 12 (white) + 18 (black) = 30 pearls.

2. **Calculate the Probability of Drawing a White Pearl First:**
- The probability of drawing a white pearl first (P(W1)) is the number of white pearls divided by the total number of pearls.
\[ P(W1) = \frac{12}{30} = \frac{2}{5} \]

3. **Update the Total Number of Pearls After the First Draw:**
- After drawing one white pearl, the total number of pearls left in the bag is 30 - 1 = 29 pearls.
- The number of black pearls remains the same at 18.

4. **Calculate the Probability of Drawing a Black Pearl Second:**
- The probability of drawing a black pearl second (P(B2)) is the number of black pearls divided by the new total number of pearls.
\[ P(B2) = \frac{18}{29} \]

5. **Calculate the Combined Probability:**
- Since the two events are independent (the second draw depends on the first draw without replacement), we multiply the probabilities of the two events:
\[ P(W1 \text{ and } B2) = P(W1) \times P(B2) = \frac{2}{5} \times \frac{18}{29} \]

6. **Perform the Multiplication:**
\[ P(W1 \text{ and } B2) = \frac{2 \times 18}{5 \times 29} = \frac{36}{145} \]

### Final Answer:
The probability that the first pearl is white and the second pearl is black is \(\frac{36}{145}\).