When rolling a pair of fair six-sided dice, each die has an equal chance of landing on any of the numbers from 1 to 6. To find the probability of different outcomes, we first need to consider the total number of possible outcomes when rolling two dice.
The total number of outcomes is calculated by multiplying the number of faces of the first die by the number of faces of the second die, which is:
Total Outcomes = 6 (first die) × 6 (second die) = 36
Now, let’s look at the probabilities for some specific outcomes:
1. Probability of rolling a sum of 7:
To achieve a sum of 7, the following combinations are possible:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)
There are 6 combinations that result in a sum of 7. Therefore, the probability is:
P(sum of 7) = Number of favorable outcomes / Total outcomes = 6/36 = 1/6
2. Probability of rolling doubles:
Doubles occur when both dice show the same number. The possible doubles when rolling two dice are:
- (1, 1)
- (2, 2)
- (3, 3)
- (4, 4)
- (5, 5)
- (6, 6)
There are 6 different doubles. Thus, the probability of rolling doubles is:
P(doubles) = Number of favorable outcomes / Total outcomes = 6/36 = 1/6
3. Probability of rolling a sum greater than 9:
To get a sum greater than 9, the following combinations are possible:
- (4, 6)
- (5, 5)
- (5, 6)
- (6, 4)
- (6, 5)
- (6, 6)
There are 6 combinations that yield a sum greater than 9. Therefore, the probability is:
P(sum > 9) = Number of favorable outcomes / Total outcomes = 6/36 = 1/6
By understanding the total combinations and specific favorable outcomes, you can calculate various probabilities when rolling a pair of fair dice.