Question

You roll two fair dice and the sum of the dice are determined. What is the probability that thesum of the dice is greater than 7? Record the answer as a fraction.

Answer 1

We have two dice and we have to find the probability that the sum of the dice is greater than 7.

We can study the sample space.

If the first dice is 1, then there is no chance that the sum of the dice is greater than 7, as the maximum value of the second dice is 6 and the sum will be 7, not greater than 7.

If the first dice is 2, then if the second dice is 6, the sum is greater than 7. We can calculate the probability of getting this combination as:

`P(x_1=2)\cdot P(x_2=6)=(1)/(6)\cdot(1)/(6)=(1)/(36)`

If the first dice is 3, then we need a 5 or a 6 in the second dice. We can calculate this probability as:

`P_{}(x_1=3)\cdot P(x_2=5,6)=(1)/(6)\cdot(2)/(6)=(2)/(36)`

If the first dice is 4, then we need a 4, 5 or 6 in the second dice. We can calculate this probability as:

`P_{}(x_1=4)\cdot P(x_2=4,5,6)=(1)/(6)\cdot(3)/(6)=(3)/(36)`

If the first dice is 5, we need at least a 3 in the second dice (could be 3, 4, 5 or 6), so we calculate this probability as:

`P_{}(x_1=5)\cdot P(x_2=3,4,5,6)=(1)/(6)\cdot(4)/(6)=(4)/(36)`

Finally, if we get a 6 in the first dice, we need at least a 2 in the second dice:

`P_{}(x_1=6)\cdot P(x_2=2,3,4,5,6)=(1)/(6)\cdot(5)/(6)=(5)/(36)`

We can now add all this probabilities and calculate the total probability of having a sum of two dice greater than 7:

`P(X>7)=0+(1)/(36)+(2)/(36)+(3)/(36)+(4)/(36)+(5)/(36)=(15)/(36)=(5)/(12)\approx0.417`

Answer: the probability of having a sum greater than 7 is P=5/12 or approximately 0.417.