Question

Find the probability that when a pair of dice are thrown, the sum of the two up faces is greater than 7 or the same number appears on each face.

Short answer

The probability that the sum of the up faces is greater than 7 or the same number appears on each face when rolling a pair of dice is \(\frac{23}{36}\).

Step-by-step solution

List the possible outcomes

A pair of dice has 6 faces each, so there are 6 x 6 = 36 possible outcomes when throwing a pair of dice. We can list these outcomes in a 6x6 table.

Calculate the probability of the first event

The first event is that the sum of the up faces is greater than 7. To find the probability of this event, first count the number of outcomes that meet this condition. In the 6x6 table, we have: - 6 outcomes with a sum of 8 - 5 outcomes with a sum of 9 - 4 outcomes with a sum of 10 - 3 outcomes with a sum of 11 - 2 outcomes with a sum of 12 So there are 6+5+4+3+2 = 20 favourable outcomes for this event. The probability of getting a sum greater than 7 is the number of favourable outcomes divided by the total number of possible outcomes: P(sum > 7) = 20/36

Calculate the probability of the second event

The second event is rolling doubles, which means getting the same number on both dice. There are 6 possible double rolls (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). There are a total of 36 possible outcomes, so the probability of rolling doubles is: P(doubles) = 6/36

Calculate the probability of both events happening at the same time

In this step, we need to find the probability of both events happening at the same time, i.e., the intersection of the two events. We can see that the doubles (4,4), (5,5), and (6,6) also have a sum greater than 7. So there are 3 favourable outcomes for the intersection of the two events. Thus, the probability of getting both events at the same time is: P(sum > 7 and doubles) = 3/36

Calculate the overall probability using the formula for the union of two events

The formula for the probability of the union of two events (A and B) is: P(A or B) = P(A) + P(B) - P(A and B) In our problem, we have: P(A or B) = P(sum > 7 or doubles) Using the probabilities found in Steps 2, 3, and 4, we can calculate the overall probability: P(sum > 7 or doubles) = P(sum > 7) + P(doubles) - P(sum > 7 and doubles) P(sum > 7 or doubles) = (20/36) + (6/36) - (3/36) P(sum > 7 or doubles) = 23/36 So the probability that the sum of the up faces is greater than 7 or the same number appears on each face when rolling a pair of dice is 23/36.