An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability that the first draw gives all white balls and second draw gives all black balls is

The correct option is 3/715.

Step-by-Step Explanation:

1. Understand the Initial Composition of the Urn:
The urn initially contains:
• White balls = 6
• Black balls = 9
• Total number of balls = 6 + 9 = 15

2. Probability of the First Draw (All White Balls):
We draw 4 balls from the urn. The probability that all 4 balls are white is the number of ways to choose 4 white balls out of 6 divided by the total number of ways to choose 4 balls out of 15:

$P(\text{First draw gives all white})=\frac{C_4^6}{C_4^{15}}$

3. Probability of the Second Draw (All Black Balls):
Since the draws are made without replacement, the 4 white balls drawn in the first step are not returned to the urn.
The composition of the urn before the second draw is:
• Remaining white balls = 6 − 4 = 2
• Remaining black balls = 9 (since no black balls were removed)
• Total remaining balls = 15 − 4 = 11

Now, we draw 4 balls from the remaining 11 balls. The probability that all 4 balls in this second draw are black is:

$P(\text{Second draw gives all black}|\text{First draw gives all white})=\frac{C_4^9}{C_4^{11}}$

4. Calculate the Combined Probability:
The overall probability P is the product of the two probabilities:

$P=\frac{C_4^6}{C_4^{15}}\times \frac{C_4^9}{C_4^{11}}$

As shown in the calculations from the image:

$P=\frac{15\times 24}{15\times 14\times 13\times 12}\times \frac{9\times 8\times 7\times 6\times 24}{24\times 11\times 10\times 9\times 8}$

5. Simplify the Fractions:
First, simplify the terms individually:
• For the first fraction:

$\frac{15\times 24}{15\times 14\times 13\times 12}=\frac{24}{14\times 13\times 12}=\frac{2}{14\times 13}=\frac{1}{7\times 13}$

• For the second fraction:

$\frac{9\times 8\times 7\times 6\times 24}{24\times 11\times 10\times 9\times 8}=\frac{7\times 6}{10\times 11}$

Now, multiply the two simplified fractions together:

$P=\frac{1}{13\times 7}\times \frac{7\times 6}{10\times 11}$

Canceling out the common factor of 7 from the numerator and denominator gives:

$P=\frac{6}{13\times 10\times 11}$

Simplifying $\frac{6}{10}$ to $\frac{3}{5}$:

$P=\frac{3}{13\times 5\times 11}=\frac{3}{715}$