Question

if two fair dice are rolled, find the probability that the sum of the dice is 8, given that the sum is greater than 6.
the probability is
(simplify your answer. type an integer or a simplified fraction.)

Explanation:

Step1: Find total outcomes

When two dice are rolled, the total number of outcomes is $6\times6 = 36$.

Step2: Find number of outcomes where sum > 6

The possible sums and their number of combinations:

• Sum = 7: Combinations are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - 6 combinations.
• Sum = 8: Combinations are (2,6), (3,5), (4,4), (5,3), (6,2) - 5 combinations.
• Sum = 9: Combinations are (3,6), (4,5), (5,4), (6,3) - 4 combinations.
• Sum = 10: Combinations are (4,6), (5,5), (6,4) - 3 combinations.
• Sum = 11: Combinations are (5,6), (6,5) - 2 combinations.
• Sum = 12: Combination is (6,6) - 1 combination.

The number of outcomes where the sum is greater than 6 is $6 + 5+4 + 3+2 + 1=21$.

Step3: Find number of outcomes where sum = 8 and sum > 6

The number of outcomes where the sum is 8 is 5.

Step4: Use conditional - probability formula

The formula for conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In the case of equally - likely outcomes, $P(A|B)=\frac{n(A\cap B)}{n(B)}$, where $n(A\cap B)$ is the number of outcomes in both $A$ and $B$, and $n(B)$ is the number of outcomes in $B$. Here, $A$ is the event that the sum is 8 and $B$ is the event that the sum is greater than 6. So $P=\frac{5}{21}$.

Answer:

$\frac{5}{21}$