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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October 10th, 2009 at 8:46 pm

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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.

October 11th, 2009 at 2:26 am

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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