A pair of dice is rolled. What is the probability that the sum showing on a pair of dice will be greater than 5 given that the first die shows a 1 ? The probability that the sum showing on a pair of dice will be greater than 5 is (Type an integer or a simplified fraction.)

Let's denote the event of interest as \( S \), where the sum of the dice is greater than 5, and let \( A \) be the event that the first die shows a 1. Since the first die shows a 1, the outcome of the first die is fixed at 1. Therefore, the only uncertainty comes from the second die, which can land on any number from 1 to 6 with equal probability. Given this, the total number of possible outcomes for the second die is 6. Now, under the condition that the first die is 1, the sum of the dice is: \[ 1 + X, \] where \( X \) is the number on the second die. We want: \[ 1 + X > 5. \] Subtract 1 from both sides: \[ X > 4. \] The possible values for \( X \) that satisfy this inequality are 5 and 6. Thus, there are 2 favorable outcomes. The probability that the sum is greater than 5 given that the first die shows a 1 is: \[ P(S \mid A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{6} = \frac{1}{3}. \] So, the probability is: \[ \frac{1}{3}. \]