To find the conditional probability that the sum of the numbers rolled on two 6-sided dice is less than 8, given that the sum is even, we can follow a step-by-step approach:
1. Total Number of Outcomes:
- When rolling two 6-sided dice, each die has [tex]\(6\)[/tex] faces. Therefore, the total number of possible outcomes is [tex]\(6 \times 6 = 36\)[/tex].
2. Favorable Outcomes for Even Sum:
- We need to identify the outcomes where the sum of the two dice is even. The sums that are even within this range (2 to 12) are [tex]\(2, 4, 6, 8, 10, 12\)[/tex].
- Listing all pairs that result in an even sum:
- Sum of [tex]\(2\)[/tex]: (1, 1)
- Sum of [tex]\(4\)[/tex]: (1, 3), (2, 2), (3, 1)
- Sum of [tex]\(6\)[/tex]: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)
- Sum of [tex]\(8\)[/tex]: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
- Sum of [tex]\(10\)[/tex]: (4, 6), (5, 5), (6, 4)
- Sum of [tex]\(12\)[/tex]: (6, 6)
- Counting these outcomes, we have 18 pairs where the sum is even.
3. Favorable Outcomes for Even Sum and Sum Less Than 8:
- Now, we identify the pairs where the sum is less than 8 and even. These sums are [tex]\(2, 4, 6\)[/tex]:
- Sum of [tex]\(2\)[/tex]: (1, 1)
- Sum of [tex]\(4\)[/tex]: (1, 3), (2, 2), (3, 1)
- Sum of [tex]\(6\)[/tex]: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)
- Counting these outcomes, we have 9 pairs where the sum is even and less than 8.
4. Conditional Probability Calculation:
- The conditional probability [tex]\(P(A|B)\)[/tex] is calculated as the number of favorable outcomes for both events divided by the number of favorable outcomes for the given condition.
- [tex]\(P(\text{Sum} < 8 \mid \text{Sum is even}) = \frac{\text{Number of outcomes with even sum and less than 8}}{\text{Number of outcomes with even sum}}\)[/tex].
- Thus, [tex]\(P(\text{Sum} < 8 \mid \text{Sum is even}) = \frac{9}{18} = 0.5\)[/tex].
Therefore, the conditional probability that the sum is less than 8 given that the sum is even is [tex]\(0.5\)[/tex].
