A fair die is thrown twice, what is the probability that the product of the outcome is a perfect square?
what are the perfect squares less than 12?
1, 4, 9
To find the probability that the product of the outcomes is a perfect square when a fair die is thrown twice, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Step 1: Determine the total number of possible outcomes.
When a fair die is thrown twice, since each throw has 6 possible outcomes (numbers 1 to 6), the total number of possible outcomes is 6 * 6 = 36.
Step 2: Determine the number of favorable outcomes.
For the product of two numbers to be a perfect square, both numbers themselves must be perfect squares. There are six perfect squares from 1 to 6: 1, 4, 9, 16, 25, and 36.
Let's consider each perfect square separately:
1 x 1: There is only one way to get a product of 1 (1 x 1).
2 x 2: There is only one way to get a product of 4 (2 x 2).
3 x 3: There is only one way to get a product of 9 (3 x 3).
4 x 4: There is only one way to get a product of 16 (4 x 4).
5 x 5: There is only one way to get a product of 25 (5 x 5).
6 x 6: There is only one way to get a product of 36 (6 x 6).
So, there are 6 favorable outcomes.
Step 3: Calculate the probability.
Now that we have the number of favorable outcomes (6) and the total number of possible outcomes (36), we can calculate the probability:
Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
Probability = 6 / 36
Probability = 1 / 6
Therefore, the probability that the product of the outcomes is a perfect square when a fair die is thrown twice is 1/6.