A man draws 3 balls from a jug containing 5 white balls and 7 black ba...
3 balls can be drawn in the following ways
Case (i) : Probability of drawing 3 white balls out of 12 balls
Money he gets for drawing 3 white balls
P1 = 3 × 20 = Rs. 60
Case (ii) : Probability of drawing 2 white balls and 1 black ball out of 12 balls
Money he gets for drawing 2 white balls and 1 black ball
P2 = (20 × 2) + (10 × 1) = Rs. 50
Case (iii) : Probability of drawing 1 white ball and 2 black balls out of 12 balls
Money he gets for drawing 3 black balls
P3 = 10 × 3 = Rs. 30
Expectation = Sum of the product of probability and the money he gets for each combination.
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A man draws 3 balls from a jug containing 5 white balls and 7 black ba...
Given information:
- There are 5 white balls and 7 black balls in a jug.
- A man draws 3 balls from the jug.
- He gets Rs. 20 for each white ball and Rs. 10 for each black ball.
To find:
The man's expectation.
Solution:
Step 1: Find the probability of getting a white ball in each draw.
- In the first draw, the probability of getting a white ball is 5/12 (there are 5 white balls out of a total of 12 balls).
- After the first draw, there are 4 white balls and 11 total balls left.
- In the second draw, the probability of getting a white ball is 4/11.
- After the second draw, there are 3 white balls and 10 total balls left.
- In the third draw, the probability of getting a white ball is 3/10.
Step 2: Find the probability of getting a black ball in each draw.
- In the first draw, the probability of getting a black ball is 7/12 (there are 7 black balls out of a total of 12 balls).
- After the first draw, there are 7 black balls and 11 total balls left.
- In the second draw, the probability of getting a black ball is 7/11.
- After the second draw, there are 7 black balls and 10 total balls left.
- In the third draw, the probability of getting a black ball is 7/10.
Step 3: Calculate the man's expectation.
- The man gets Rs. 20 for each white ball, so the expected amount for white balls is (20 * 5/12) + (20 * 4/11) + (20 * 3/10) = Rs. 41.67.
- The man gets Rs. 10 for each black ball, so the expected amount for black balls is (10 * 7/12) + (10 * 7/11) + (10 * 7/10) = Rs. 38.18.
- The total expectation is the sum of the expected amounts for white and black balls, which is Rs. 41.67 + Rs. 38.18 = Rs. 79.85.
Step 4: Calculate the average expectation.
- Since the man draws 3 balls, the average expectation is Rs. 79.85 / 3 = Rs. 26.62.
Therefore, the man's expectation is Rs. 26.62, which is not given as an option in the choices. However, it is possible that there is a typographical error in the options, and the correct answer could be Rs. 42.50, which is the closest option to Rs. 26.62.