Question

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

• A. \( \frac{3}{4} \)
• B. \( \frac{1}{4} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{2}{3} \)
• E. \( \frac{1}{16} \)

Hint:

In probability, whenever a condition says "at least one" or "even product," always check if calculating the "none" or "odd" case is easier. Subtracting from 1 often saves significant time compared to tallying all favorable cases.

Answer 1

The Correct Option is A

Concept: When two dice are thrown, the total number of outcomes is \( 6 \times 6 = 36 \). The product of two numbers is even if at least one of the numbers is even. Conversely, the product is odd only if both numbers are odd. Using the complement rule \( P(\text{Even}) = 1 - P(\text{Odd}) \) is much faster.

Step 1: Calculate the probability of the product being odd.
For the product of two numbers to be odd, both numbers must be odd. The odd numbers on a die are {1, 3, 5} (3 choices).
• Number of ways to get two odd numbers = \( 3 \times 3 = 9 \).
• \( P(\text{Odd Product}) = \frac{9}{36} = \frac{1}{4} \).

Step 2: Calculate the probability of the product being even.
Using the complement: \[ P(\text{Even Product}) = 1 - P(\text{Odd Product}) \] \[ P = 1 - \frac{1}{4} = \frac{3}{4} \]