Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even ?

### Concept & Strategy The product of two numbers is even if at least one of the numbers is even. The only way a product can be odd is if *both* numbers are odd. Using the complement rule is much faster here: $P(\text{Even Product}) = 1 - P(\text{Odd Product})$. $$P(E) = 1 - P(E')$$ ### Step-by-Step Solution 1. **Identify Sample Space:** For two dice, total outcomes $n(S) = 36$. 2. **Find Complement (Odd Product):** A product is odd only when both dice show an odd number (1, 3, or 5). The number of odd outcomes on one die is 3. Total odd pairs = $3 \times 3 = 9$. So, $n(\text{Odd Product}) = 9$. 3. **Calculate Probability of Complement:** $P(\text{Odd Product}) = \frac{9}{36} = \frac{1}{4}$ 4. **Calculate Final Probability:** $P(\text{Even Product}) = 1 - P(\text{Odd Product}) = 1 - \frac{1}{4} = \frac{3}{4}$ ### Exam Strategy & Shortcut Instead of listing all 27 pairs that produce an even product, recognizing that $Odd \times Odd = Odd$ allows you to immediately see that $1/2 \times 1/2 = 1/4$ of the outcomes are odd. The rest must be even: $1 - 1/4 = 3/4$. ### Common Pitfall Attempting to list out every single combination that gives an even product takes too much time and heavily increases the risk of miscounting. ### Final Answer Therefore, the correct answer is **$\frac{3}{4}$**.