Paradox Probability


This is a famous paradox which has caused a great deal of argument and disbelief from many who cannot accept the correct answer. Four balls are placed in a hat. One is white, one is blue and the other two are red. The bag is shaken and someone draws two balls from the hat. He looks at the two balls and announces that at least one of them is red. What are the chances that the other ball he has drawn out is also red?

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    The key to understanding the paradox is to consider the probability of each possible outcome before the second ball is drawn.

    At the beginning, when all four balls are in the hat, the probability of drawing the white ball is 1/4, the probability of drawing the blue ball is 1/4, and the probability of drawing one of the red balls is 2/4 or 1/2.

    When the first ball is drawn and revealed to be red, the probability of the remaining balls in the hat changes. There are now three balls left, one white, one blue, and one red. The probability of drawing the white or blue ball is 1/3 each, and the probability of drawing the remaining red ball is 1/3 as well.

    Given that at least one of the balls drawn is red, we know that the red-red outcome is possible, and so is either the red-blue or red-white outcome.

    Therefore, the probability that the second ball drawn is also red is the probability of the red-red outcome divided by the probability of the two possible outcomes that include at least one red ball, which is 1/2. So the probability of the second ball being red is 1/2, or 50%, which is different from what intuition might suggest.

    Moshe Expert Answered on 8th March 2023.
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