Find the probability that sum of the numbers showing on the two dice is 8, given that atleast one dice doesn't show five.

To find the probability that the sum of the numbers showing on two dice is 8, given that at least one die does not show a five, we can break this problem down into manageable parts. First, we need to determine the total number of outcomes where the sum is 8, and then we will consider the restriction regarding the number five. Let's go through this step by step.

Step 1: Total Outcomes for Two Dice

When rolling two six-sided dice, there are a total of 36 possible outcomes, since each die has 6 faces. This is calculated as:

• 6 options for the first die
• 6 options for the second die

Thus, the total outcomes = 6 × 6 = 36.

Step 2: Outcomes Where the Sum is 8

Next, we need to identify the combinations of the two dice that result in a sum of 8. The pairs that achieve this are:

• (2, 6)
• (3, 5)
• (4, 4)
• (5, 3)
• (6, 2)

Counting these, we find there are 5 outcomes where the sum equals 8.

Step 3: Exclude Outcomes with a Five

Now, we need to consider the condition that at least one die does not show a five. We will exclude any outcomes from our previous list that contain a five:

• (3, 5)
• (5, 3)

Removing these two pairs leaves us with the following valid outcomes for a sum of 8:

• (2, 6)
• (4, 4)
• (6, 2)

So, there are 3 outcomes that meet both criteria: the sum is 8, and at least one die does not show a five.

Step 4: Total Valid Outcomes Under the Condition

Next, we need to find the total number of outcomes that satisfy the condition of having at least one die not showing a five. The total outcomes that include at least one five can be calculated as follows:

• Outcomes with at least one five: (1,5), (2,5), (3,5), (4,5), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,5) — totaling 11 outcomes.

Thus, the outcomes that do not include a five are:

• Total outcomes (36) - Outcomes with at least one five (11) = 36 - 11 = 25.

Step 5: Calculate the Probability

The probability we are looking for is the ratio of the number of favorable outcomes (sum is 8 and at least one die does not show five) to the total number of valid outcomes (at least one die does not show five).

So, the probability is:

P(sum = 8 | at least one die ≠ 5) = Number of favorable outcomes / Total valid outcomes

Substituting the values we found:

P(sum = 8 | at least one die ≠ 5) = 3 / 25

Final Result

Therefore, the probability that the sum of the numbers showing on the two dice is 8, given that at least one die does not show a five, is 3/25.