A pair of dice is rolled. What is the probability that the sum showing on a pair of dice well be greater than 5 given that the first die shows a 2 ? The probability that the sum showing on a pair of dice will be greater than 5 is (Type an integer or a simplified fraction.)

When the first die shows a 2, the second die can show any number from 1 to 6. To find the outcomes where the sum is greater than 5, we can list them: (2,4), (2,5), and (2,6) which sum to 6, 7, and 8 respectively. This gives us a total of 3 favorable outcomes. Now, there are 6 possible outcomes for the second die (1, 2, 3, 4, 5, 6). Therefore, the probability that the sum is greater than 5 given the first die shows a 2 is: \[ P(\text{Sum} > 5 | \text{First die} = 2) = \frac{3}{6} = \frac{1}{2} \] So the answer is \( \frac{1}{2} \). In the world of probability, every roll of a die showcases a little piece of randomness, making it akin to rolling a tiny piece of the universe! Fun fact: the average roll of two six-sided dice is 7, a result that keeps probability enthusiasts quizzing over various strategies. If you ever get stuck with probability puzzles, it can help to lay out all possible outcomes. Organizing them in a table or a tree diagram showcases the relationships between variables. Avoid overthinking when all you need is a clear visual to see the answer right in front of you!