A pair of dice is thrown twice. What is the probability that the faces of the second throw are the same as the first?
Two types of outcomes are possible when you throw a pair of dice. In the first case, both dice show the same number (11, 22, 33, 44, 55, 66). In the second type, they differ (12, 13, etc.). We solve both scenarios separately.
Different
For a pair of dice, there are a total of 6 x 6 = 36 events possible. Of these, 6 are the same type (11, 22, 33, 44, 55, 66), and 36 – 6 = 30 are different. The probability of different numbers is 30/36.
When you repeat the throw, the exact two numbers come only in 2 out of the 36 outcomes. E.g., if 12 comes first, there are two ways of getting 12 in the second – 1 from the first die, 2 from the second OR 2 from the first or 1 from the second. The probability is 2/36.
The overall possibility of a repeat of pairs is (30/36)x(2/36)
Same
We have already seen that the probability of getting the same numbers in the first pair is 6 out of 36 (6/36). In the repeat, the situation only happens 1 in 36 (1/36).
The overall possibility of a repeat is (6/36)x(1/36)
We need to add the two probabilities to get the probability of getting the same in the repeat.
(30/36)x(2/36) + (6/36)x(1/36) = 0.0509; a 5% chance.
