Five Same Numbers

You roll five fair, six-sided dice, all at the same time. In that one roll, what is the probability of getting five of the same number?
Source: NCTM Mathematics Teacher, September 2006

Solution
When you roll five dice at the same time, 6 possible numbers from 1 through 6 can come up on each of the five dice. The number of possible outcomes = 6\times 6\times 6\times 6\times 6=6^5. There are 6 possible ways the five dice come up with the same number: 11111,22222,33333,44444,55555,66666. The probability of getting five of the same number in one roll = 6/6^5=1/6^4=1/1296.

Answer: 1/1296

Alternative solution
Suppose the event 11111 occurs on a roll. Since the number 1 on the first dice is independent of the number 1 on the second dice, etc., the probability of the event 11111=1/6\times 1/6\times 1/6\times 1/6\times 1/6=1/6^5. The probability of getting five of the same number in one roll = 1/6^5+1/6^5+1/6^5+1/6^5+1/6^5=6(1/6^5)=1/6^4=1/1296.

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About mvtrinh

Retired high school math teacher.
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