Two fair six faced dice are thrown simultaneously. What is the probability that the product of the two numbers obtained on the upper faces is an even number?

Introduction / Context:
This question deals with the outcome of rolling two standard dice and examines the parity of the product of the two numbers. Instead of counting all even products directly, it is usually simpler to consider when the product is odd and then take the complement. This illustrates how complementary events can make calculations easier in basic probability problems.

Given Data / Assumptions:

• Two fair six sided dice are rolled simultaneously.
• Each die can show any integer from 1 to 6.
• All 36 ordered pairs of outcomes are equally likely.
• We need the probability that the product of the two numbers is even.

Concept / Approach:
A product of two integers is odd only when both integers are odd. If either number is even, then the product is even. Hence, the easiest approach is to first find the probability that both numbers are odd and therefore the product is odd, and then subtract that probability from 1 to obtain the probability that the product is even. This uses the complement rule in probability.

Step-by-Step Solution:
Step 1: Each die has 3 odd faces, which are 1, 3 and 5, and 3 even faces, which are 2, 4 and 6.Step 2: The total number of outcomes when two dice are rolled is 6 * 6 = 36.Step 3: Count outcomes where both dice show odd numbers. There are 3 odd choices for the first die and 3 odd choices for the second die, giving 3 * 3 = 9 outcomes.Step 4: In these 9 outcomes, the product is odd. Therefore, the probability that the product is odd is 9 / 36.Step 5: Probability that the product is even is the complement: 1 minus 9 / 36 = 27 / 36.Step 6: Simplify 27 / 36 by dividing numerator and denominator by 9 to get 3 / 4.
Verification / Alternative check:
As an alternative, we can count even products directly. If either die is even, the product is even. We can count all combinations where at least one die is even and show that there are 27 such outcomes. Another way is to check a few specific examples like (2,1), (4,3) and (6,5) to confirm that the presence of at least one even number guarantees an even product. Both approaches reach the same probability value, which validates the result.

Why Other Options Are Wrong:
3/8 and 5/16 are too small and would imply that the majority of products are odd, which is not reasonable given that half of the faces on each die are even. 2/7 does not simplify nicely in relation to powers of 6 and does not match any clear counting logic. 1/4 is the probability that both numbers are even only under certain miscalculations, not the probability that the product is even in general.

Common Pitfalls:
A common mistake is to suppose that even and odd products occur equally often and to guess 1/2, but the calculations show that even products are more frequent. Another error is miscounting the number of odd faces on each die or forgetting that all 36 ordered pairs are equally likely. Remembering that the complement of an event often simplifies counting is a key lesson from this problem.

Final Answer:
The probability that the product of the two numbers is even is 3/4.