2. Given that a die is rolled twice and the sum of the numbers is noted to be 6, what is the conditional probability that the number 4 has occurred at least once?

To find the conditional probability that the number 4 has occurred at least once, given that the sum of the numbers rolled is 6, we need to consider all the possible outcomes that satisfy this condition.

Step 1: Determine the sample space When a die is rolled twice, the sample space consists of all possible outcomes. Each roll has 6 possible outcomes, so the sample space contains 6 x 6 = 36 possible outcomes.

Step 2: Identify the outcomes that satisfy the condition To find the outcomes where the sum of the numbers rolled is 6, we can list all the possible combinations: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)

Step 3: Determine the outcomes where the number 4 has occurred at least once From the list of outcomes above, we can see that the number 4 occurs in the following combinations: (2, 4), (4, 2)

Step 4: Calculate the conditional probability The conditional probability is the number of outcomes where the number 4 has occurred at least once, divided by the total number of outcomes satisfying the condition.

In this case, there are 2 outcomes where the number 4 has occurred at least once, and 5 outcomes in total that satisfy the condition (as listed in Step 2).

Therefore, the conditional probability that the number 4 has occurred at least once, given that the sum of the numbers rolled is 6, is 2/5 or 0.4.