An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black ?

Question

Accepted Answer

To determine the probability that both drawn balls are black, we need to consider the process of drawing two balls one after the other without replacement from the urn, which initially contains 10 black balls and 5 white balls, making a total of 15 balls.

1. **Probability of the first ball being black:**
   - There are 10 black balls out of a total of 15 balls.
   - Therefore, the probability of drawing a black ball first is $ \frac{10}{15} $.

2. **Probability of the second ball being black given the first was black:**
   - After drawing one black ball, there are now 9 black balls left and the total number of balls is reduced to 14.
   - Therefore, the probability of drawing a second black ball is $ \frac{9}{14} $.

3. **Combined probability for both events:**
   - Since these are independent events (the outcome of the first draw affects the second), we multiply the probabilities of each event:
   $$
   \text{Probability of both balls being black} = \frac{10}{15} \times \frac{9}{14} = \frac{90}{210} = \frac{3}{7}
   $$

Thus, the probability that both drawn balls are black is $ \frac{3}{7} $, which corresponds to Option 2.

**Explanation for why other options are incorrect:**

- **Option 1: $\frac{2}{7}$**: This option is incorrect because the calculation of the probability does not yield $\frac{2}{7}$. The correct calculation, as shown above, results in $\frac{3}{7}$.

- **Option 3: $\frac{4}{7}$**: This option is incorrect because it overestimates the probability. The correct calculation, based on the number of black balls and the total number of balls, results in $\frac{3}{7}$.

- **Option 4: $\frac{5}{7}$**: This option is incorrect because it significantly overestimates the probability. The correct calculation, considering the reduction in the number of black balls after the first draw, results in $\frac{3}{7}$.

In conclusion, the correct probability that both drawn balls are black is $\frac{3}{7}$, which is why Option 2 is the correct answer.