A box contains 10 balls numbered 1 to 10. Two balls are drawn with replacement. Find the probability that the product of the numbers on the two balls is a perfect square.
Answer
9/50
The correct answer is 9/50 — Probability = 18/100 =9/50.
Step-by-step solution
- Total ordered pairs when drawing two balls with replacement = 10*10 =100.
- For a product to be a perfect square, both numbers must share all prime factors with even exponents.
- Enumerate all ordered pairs (a,b) with a,b∈{1,…,10} whose product is a square: (1,1) (1,4),(4,1) (2,2) (1,9),(9,1) (3,3) (2,8),(8,2) (4,4) (5,5) (4,9),(9,4) (6,6) (7,7) (8,8) (9,9) (10,10) Count =18 pairs.
- Probability = 18/100 =9/50.
Common mistake: Considering only equal numbers and ignoring cross pairs or not counting ordered pairs
Concept tested: Probability of perfect square product with replacement
Published · How this solution was prepared and checked
← All Probability questions