Question

An urn contains 15 black and 10 white balls. Two balls are drawn from the urn one after another without replacement. What is the probability that both drawn balls are black?

• A. \(\frac{5}{8}\)
• B. \(\frac{7}{20}\)
• C. \(\frac{6}{25}\)
• D. \(\frac{1}{5}\)

Hint:

Combinations method: $P = \frac{\binom{15}{2}}{\binom{25}{2}} = \frac{\frac{15 \times 14}{2}}{\frac{25 \times 24}{2}} = \frac{105}{300} = \frac{7}{20}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Drawing without replacement is modeled as conditional probability or through hypergeometric combinations.
Key Formula or Approach:
\[ P(B_1 \cap B_2) = P(B_1) \cdot P(B_2 | B_1) = \frac{\binom{n_{\text{black}}}{2}}{\binom{n_{\text{total}}}{2}} \]

Step 2: Detailed Explanation:
Total number of balls initially:
\[ N = 15 \text{ (black)} + 10 \text{ (white)} = 25 \text{ balls} \]
Probability of drawing a black ball on the first draw:
\[ P(B_1) = \frac{15}{25} = \frac{3}{5} \]
After drawing one black ball, 14 black balls and 24 total balls remain:
\[ P(B_2 | B_1) = \frac{14}{24} = \frac{7}{12} \]
Combined probability that both balls are black:
\[ P = \frac{3}{5} \times \frac{7}{12} = \frac{21}{60} = \frac{7}{20} \]

Step 3: Final Answer:
Hence, the probability that both drawn balls are black is \(\frac{7}{20}\), matching option (B).