Final answer:
The probability that exactly two dice show the same number when three dice are rolled is 5/12. This is calculated by determining the number of favorable outcomes (90) and dividing it by the total number of outcomes (216). Correct option is C. 5/12
Step-by-step explanation:
When 3 dice are rolled simultaneously, the probability that exactly two of the dice will come up as the same number can be calculated by first determining the total number of possible outcomes when three dice are rolled. Each die has 6 faces, therefore the total number of outcomes for three dice is 6 x 6 x 6, which equals 216. To have exactly two dice showing the same number, we follow these steps:
- Choose which two dice will show the same number. There are 3 ways to do this because the pair can be the first and second dice, the second and third dice, or the first and third dice.
- Choose the number that will appear on both dice. There are 6 choices for this step (one for each number on a die).
- Choose the number for the remaining die. Since it cannot be the same as the pair, there are 5 choices for this step.
- Multiply the number of choices for each of the steps. The total number of favorable outcomes is 3 (ways to choose the pair) x 6 (choices for the pair's number) x 5 (choices for the remaining die's number), resulting in 90 favorable outcomes.
To find the probability, we divide the number of favorable outcomes by the total number of outcomes: 90/216, which simplifies to 5/12 (or 15/36 when simplified to 60/216). Therefore, the correct answer is C. 5/12.
It's important to note that with probability questions involving dice, we typically calculate outcomes based on the product rule and the number of favorable versus possible outcomes to come up with the probability.