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Suppose that you have an urn with 20 red balls and 40 blue balls. You pull out 15 balls without putting them back. On average, how many of the 15 balls will be red?

Answer :

On average, out of 15 balls drawn without replacement from an urn containing 20 red balls and 40 blue balls, 5 balls are expected to be red. This is because 1/3 of the balls in the urn are red, and 1/3 of the 15 balls drawn would be red.

The question is asking for the expected number of red balls one would draw when selecting 15 balls from an urn containing 20 red balls and 40 blue balls, without replacement. To find this, you can use the concept of expectation in probability. The expectation is the sum of all possible outcomes weighted by their probabilities. In this case, the probability of drawing a red ball each time is the number of red balls remaining divided by the total number of balls remaining.

To get the expected number of red balls out of the 15 drawn, multiply the probability of drawing a red ball each time by the number of draws. The probability starts at 20/60 for the first ball and decreases as red balls are removed and the total number of balls in the urn decreases. The expected number is the sum of the probabilities for drawing a red ball each time over the 15 draws. However, it's simpler to calculate by recognizing the proportion of red balls in the urn and multiplying by the number of draws. Since 1/3 of the balls are red (20 out of 60), we expect 1/3 of the 15 balls drawn to be red, which would be 5. Therefore, on average, 5 of the 15 balls will be red.

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