# Sum obtained by rolling all 6 dice

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*

## Answer

In order to calculate the expected value of the sum obtained by rolling all six dice, we can analyze the probability of each number appearing on the up face and multiply it by the value of that number. Then, we can sum these products to find the expected value.

Since there is only one face with a number on each die, the probability of each number from 1 to 6 appearing on a die is 1/6. The probability of a blank face showing is 5/6.

Let’s find the expected value for each die and then multiply it by 6 (since there are 6 dice).

For each die:

- The expected value for the number 1 is (1/6) * 1 = 1/6
- The expected value for the number 2 is (1/6) * 2 = 2/6
- The expected value for the number 3 is (1/6) * 3 = 3/6
- The expected value for the number 4 is (1/6) * 4 = 4/6
- The expected value for the number 5 is (1/6) * 5 = 5/6
- The expected value for the number 6 is (1/6) * 6 = 6/6

The sum of these expected values is (1/6) + (2/6) + (3/6) + (4/6) + (5/6) + (6/6) = 21/6.

Now, since there are 6 dice, we multiply the expected value for one die by 6:

Expected value of the sum obtained by rolling all 6 dice = 6 * (21/6) = 21.

So, the expected value of the sum obtained by rolling all six dice is 21.