To find the probability of drawing a white ball from urn u2 after transferring balls from urn u1, we need to analyze the two scenarios based on the coin toss.
Scenario Analysis
1. Coin Toss Results in Heads
If the coin shows heads, we transfer 1 ball from u1 to u2. The possible outcomes are:
- Transfer a white ball: Probability = 3/5
- Transfer a red ball: Probability = 2/5
After this transfer, urn u2 will have:
- If a white ball is transferred: 2 white balls (1 original + 1 transferred)
- If a red ball is transferred: 1 white ball (original)
The probabilities of drawing a white ball from u2 in this case are:
- From u2 with 2 white balls: Probability = 2/2 = 1
- From u2 with 1 white ball: Probability = 1/2
2. Coin Toss Results in Tails
If the coin shows tails, we transfer 2 balls from u1 to u2. The possible combinations are:
- 2 white balls: Probability = (3/5) * (2/4) = 3/10
- 1 white and 1 red ball: Probability = (3/5) * (2/4) + (2/5) * (3/4) = 3/10 + 3/10 = 6/10 = 3/5
- 2 red balls: Probability = (2/5) * (1/4) = 1/10
After this transfer, urn u2 will have:
- 2 white balls: 3 white balls
- 1 white and 1 red: 2 white and 1 red
- 2 red balls: 1 white and 2 red
The probabilities of drawing a white ball from u2 in this case are:
- 3 white balls: Probability = 3/3 = 1
- 2 white and 1 red: Probability = 2/3
- 1 white and 2 red: Probability = 1/3
Calculating Total Probability
Now, we combine the probabilities from both scenarios:
- Probability of heads: 1/2 * (1 * 3/5 + 1/2 * 2/5) = 1/2 * (3/5 + 1/5) = 1/2 * 4/5 = 2/5
- Probability of tails: 1/2 * (1 * 3/10 + 2/3 * 3/5 + 1/3 * 1/10) = 1/2 * (3/10 + 2/10 + 1/30) = 1/2 * (9/30 + 1/30) = 1/2 * 10/30 = 5/30
Final Probability Calculation
Adding both probabilities gives:
Probability of drawing a white ball from u2 = (2/5) + (5/30) = (12/30) + (5/30) = 17/30.
Thus, the probability of drawing a white ball from urn u2 is 19/30.
The correct answer is C) 19/30.
