Understanding the Problem
To find the probability that the product of the numbers on two unbiased dice is a perfect square, we first need to analyze the possible outcomes.
Sample Space
- Each die has 6 faces, so the total number of outcomes when throwing two dice is:
- 6 (from the first die) × 6 (from the second die) = 36 outcomes.
Perfect Squares and Their Products
- A perfect square is a number that can be expressed as the square of an integer. The perfect squares within the range of products from two dice (1 to 36) are:
- 1 (1x1), 4 (2x2), 9 (3x3), 16 (4x4), 25 (5x5), and 36 (6x6).
Finding Favorable Outcomes
- We need to find pairs (a, b) such that the product \( ab \) is a perfect square. The pairs that yield perfect square products are:
- (1,1) → 1
- (1,4) and (4,1) → 4
- (2,2) → 4
- (3,3) → 9
- (1,9) and (9,1) → 9 (not possible with dice)
- (4,4) → 16
- (6,6) → 36
- The valid pairs for perfect squares are:
- (1,1), (2,2), (3,3), (4,4), (6,6)
- Total favorable outcomes: 7 pairs.
Calculating the Probability
- The probability \( P \) that the product is a perfect square is calculated as follows:
\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{7}{36}
\]
Final Answer
Thus, the probability that the product of the numbers on two unbiased dice is a perfect square is \( \frac{7}{36} \).