Find the Probability That Two Uniform Variables Have Product Above One-Half

Let `X` and `Y` be independent random variables, each uniformly distributed on the interval `(0, 1)`. What is the probability that `X * Y > 1/2`? ### Constraints and Clarifications Use the continuous uniform distribution and the given independence. Explain the relevant region or conditional probability and give an exact expression. The distinction between strict and non-strict inequality at the boundary has probability zero here. ```hint Locate the feasible part of the square For a fixed value of `X`, determine which values of `Y` can satisfy the product inequality and when that interval is nonempty. ``` ### What a Strong Answer Covers - The uniform joint density on the unit square and the product-inequality region. - Correct integration limits rather than integrating over impossible values of `X`. - An exact probability, a reasonable numerical interpretation, and a simple bound that checks the result. ### Follow-up Questions 1. How does the expression change for `P(X * Y > t)` with `0 < t < 1`? 2. Why is multiplying `P(X > 1/2)` and `P(Y > 1/2)` not the answer to the original question?