Question

What is the conditional probability that the first die is six given that the sum of the dice is seven?

Short answer

The conditional probability that the first die is a six given that the sum of the dice is seven is \(\frac{1}{6}\).

Step-by-step solution

Determine the sample space for rolling two dice

When rolling two six-sided dice, there are a total of six possible outcomes for each die. This results in a sample space of size 6 x 6 = 36 possible outcomes (ordered pairs).

List the favourable outcomes for event A

Event A is the event that the first die shows a six. We can list these outcomes as follows: \((6,1), (6,2), (6,3), (6,4), (6,5), (6,6)\). There are 6 favourable outcomes for event A.

List the favourable outcomes for event B

Event B is the event that the sum of the dice is seven. We can list these outcomes as follows: \((1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\). There are 6 favourable outcomes for event B.

Find the favourable outcomes for both A and B events

To find the favourable outcomes for both events A and B, we look for outcomes that are in both event A and event B. This results in the outcome \((6,1)\), which is the only outcome where the first die is a six and the sum of the dice is seven.

Calculate the conditional probability P(A|B)

Now we can apply the conditional probability formula: P(A|B) = P(A and B) / P(B). First, we find P(A and B), which is the probability of the favourable outcome \((6,1)\). Since there is only one favourable outcome and there are 36 possible outcomes in total, P(A and B) = 1/36. Next, we find P(B), which is the probability of the event B (the sum of the dice is seven). There are 6 favourable outcomes for event B and 36 possible outcomes in total, so P(B) = 6/36 = 1/6. Finally, we calculate P(A|B) = P(A and B) / P(B) = (1/36) / (1/6) = (1/36) * (6/1) = 1/6. Thus, the conditional probability that the first die is a six given that the sum of the dice is seven is 1/6.