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What is the expected value of the sum obtained by rolling all 6 dice?

Early studies were conducted by the Italian mathematician Girolamo Cardona (1501-1576). One of the many dice games that Cardona studied
was played with six 6-sided dice. Each of these six dice had five blank faces and one face with a number. The numbers 1 through 6 each appeared on one of the six dice. All 6 dice were rolled at once, and the payoff to the gambler was based on the sum of the numbers showing on the up faces.

Question posted by: sweetlilac89

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2 Responses to “Sum obtained by rolling all 6 dice”

  1. Paul L Says:


    I don’t there is an expected sum, since there is only a 1 in 6 chance of you getting a number on a given die. So out of the 6 dice, you’ll probably only get one number, which will be in the 1 to 6 range.

    But if you’re talking averages, you’ll be rolling a 3 or 4 most of the time.

  2. thomasoa Says:

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    The expected sum is the sum of the expectation for each die.

    In this case, the first die has an expectation of 1/6, the second 2/6, the third 3/6, etc. So the expected value of the sum is:

    (1+2+3+4+5+6)/6 = 21/6 = 3.5

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