Problem:
Each of 6 balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other 5 balls?
Answer Choices:
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B.
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E.
Solution:
The specified event will occur if and only if there are 3 balls of each color. Indeed, if they are painted that way, then each ball has a different color than of the other balls. Conversely, if 4 or more balls are black, or 4 or more are white, then each of those balls has the same color as at least of the other balls.
The number of ways to paint the balls so that there are 3 balls of each color is the number of ways to choose 3 of the 6 balls to be white, which is . There are equally likely ways to paint the balls, so the requested probability is equal to .
The problems on this page are the property of the MAA's American Mathematics Competitions