Final answer:
When rolling two six-sided dice, the probabilities are as follows: a) dice show the same number is 1/6, b) first die showing a larger number than the second is 5/12, c) sum of the dice is even is 1/2, and d) product is a perfect square is 7/36.
Step-by-step explanation:
We are looking at the probabilities of several events when two six-sided dice are rolled. Here's a step-by-step explanation:
- a) The probability of two dice showing the same number: There are 6 possible ways this can happen (both dice showing 1, both showing 2, ..., both showing 6), out of 36 total possible outcomes (6 choices for the first die multiplied by 6 choices for the second die). Therefore, the probability is 6/36 or 1/6.
- b) The probability of the first die showing a larger number than the second: There are 15 possible outcomes for this event (2-1, 3-1, 3-2, ..., 6-5). So the probability is 15/36, which simplifies to 5/12.
- c) The probability that the sum of the dice is even: An even sum occurs when both dice are either even (even + even = even) or odd (odd + odd = even). There are 3 even and 3 odd numbers on a die, so there are 3*3 + 3*3 = 18 possible outcomes out of 36, giving a probability of 1/2.
- d) The probability that the product of the dice is a perfect square: The perfect square products from two dice are 1 (1x1), 4 (1x4 or 2x2), 9 (1x9 or 3x3), 16 (2x8 or 4x4), 25 (5x5), and 36 (6x6). Counting each pair of factors gives us 7 successful outcomes out of 36 possible outcomes, so the probability is 7/36.