Urn 1 contains 1933 white and 2067 black balls, while urn 2 contains 191 white and 167 black balls.
43 balls are randomly selected from urn 1 and are then put in urn 2. If 46 balls are then randomly
selected from urn 2, compute the expected number of white balls in the trio

To find the expected number of white balls in the trio after transferring balls from urn 1 to urn 2 and then selecting from urn 2, we need to break down the problem step by step. Let's analyze the situation carefully.

Understanding the Initial Setup

We have two urns:

• Urn 1: 1933 white balls and 2067 black balls
• Urn 2: 191 white balls and 167 black balls

Transferring Balls from Urn 1 to Urn 2

First, we randomly select 43 balls from urn 1. The total number of balls in urn 1 is:

Total in Urn 1 = 1933 + 2067 = 4000 balls

Next, we need to determine the expected number of white balls transferred. The probability of selecting a white ball from urn 1 is:

P(White from Urn 1) = Number of White Balls / Total Balls = 1933 / 4000

Now, the expected number of white balls (E) transferred can be calculated as:

E(White from Urn 1) = 43 * P(White from Urn 1) = 43 * (1933 / 4000)

Calculating the Expected Number of White Balls Transferred

Let's compute this value:

E(White from Urn 1) = 43 * (1933 / 4000) ≈ 43 * 0.48325 ≈ 20.77

Since we can’t have a fraction of a ball, we can consider this as approximately 21 white balls being transferred from urn 1 to urn 2.

Updating Urn 2's Composition

After transferring the balls, the new composition of urn 2 will be:

• White Balls in Urn 2: 191 + 21 = 212
• Black Balls in Urn 2: 167 + (43 - 21) = 189

Now, urn 2 contains:

Total in Urn 2 = 212 + 189 = 401 balls

Selecting Balls from Urn 2

Next, we randomly select 46 balls from urn 2. We want to find the expected number of white balls in this selection. The probability of selecting a white ball from urn 2 is:

P(White from Urn 2) = Number of White Balls / Total Balls = 212 / 401

The expected number of white balls (E) in the selection of 46 balls is:

E(White from Urn 2) = 46 * P(White from Urn 2) = 46 * (212 / 401)

Calculating the Final Expected Number of White Balls

Now, let's compute this value:

E(White from Urn 2) = 46 * (212 / 401) ≈ 46 * 0.52843 ≈ 24.29

Thus, the expected number of white balls in the selection of 46 balls from urn 2 is approximately 24.29, which we can round to about 24 or 25 white balls.

Final Thoughts

In summary, after transferring balls from urn 1 to urn 2 and selecting from urn 2, the expected number of white balls in the trio is approximately 24 or 25. This methodical approach allows us to use probabilities and expectations to solve complex problems involving random selections.

To tackle the problem of finding the expected number of white balls in the selection from urn 2 after transferring some balls from urn 1, we need to break it down into manageable steps. Let's analyze the situation carefully.

Understanding the Initial Setup

We have two urns:

• Urn 1: 1933 white balls and 2067 black balls.
• Urn 2: 191 white balls and 167 black balls.

Transferring Balls from Urn 1 to Urn 2

First, we randomly select 43 balls from urn 1. The total number of balls in urn 1 is:

Total in Urn 1 = 1933 + 2067 = 4000 balls

Next, we need to determine the expected number of white balls transferred to urn 2. The probability of selecting a white ball from urn 1 is:

P(White from Urn 1) = Number of White Balls / Total Balls = 1933 / 4000

Now, the expected number of white balls (E[W1]) selected from urn 1 can be calculated as:

E[W1] = 43 * P(White from Urn 1) = 43 * (1933 / 4000)

Calculating the Expected Number of White Balls

Let's compute that:

E[W1] = 43 * (1933 / 4000) ≈ 43 * 0.48325 ≈ 20.77

This means we expect approximately 20.77 white balls to be transferred from urn 1 to urn 2. Since we can't have a fraction of a ball, we can interpret this as an average over many trials.

Updating Urn 2's Composition

After transferring the balls, the new composition of urn 2 will be:

• White Balls in Urn 2: 191 + E[W1] ≈ 191 + 20.77 ≈ 211.77
• Black Balls in Urn 2: 167 + (43 - E[W1]) ≈ 167 + (43 - 20.77) ≈ 189.23

Calculating the Expected Number of White Balls in the Final Selection

Now, we need to find the expected number of white balls when we randomly select 46 balls from the updated urn 2. The total number of balls in urn 2 after the transfer is:

Total in Urn 2 = 211.77 + 189.23 = 401 balls

The probability of selecting a white ball from the updated urn 2 is:

P(White from Urn 2) = White Balls in Urn 2 / Total Balls in Urn 2 ≈ 211.77 / 401

Now, the expected number of white balls (E[W2]) in the selection of 46 balls from urn 2 is:

E[W2] = 46 * P(White from Urn 2) = 46 * (211.77 / 401)

Final Calculation

Calculating this gives:

E[W2] ≈ 46 * 0.528 ≈ 24.25

Thus, the expected number of white balls in the selection of 46 balls from urn 2 is approximately 24.25. This value represents the average outcome over many trials of this process.

Summary

In conclusion, after transferring 43 balls from urn 1 to urn 2 and then selecting 46 balls from urn 2, the expected number of white balls in that selection is about 24.25. This approach illustrates how we can use probabilities and expected values to analyze random selections and transfers in a systematic way.