Understanding the Probability of Rolling a Sum of 9 or Higher
When rolling two fair six-sided dice, we want to find the probability that the sum of the numbers on the top faces of the dice is 9 or greater. Let’s break this down step by step.
Step 1: Total Possible Outcomes
Each die has 6 sides, hence when rolling two dice, the total possible outcomes are:
6 (for the first die) × 6 (for the second die) = 36 possible outcomes.
Step 2: Favorable Outcomes
We must identify the combinations of the dice that sum to 9 or higher. Here are the possible sums and their corresponding combinations:
- Sum of 9:
- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)
Total combinations: 4
- Sum of 10:
- (4, 6)
- (5, 5)
- (6, 4)
Total combinations: 3
- Sum of 11:
- (5, 6)
- (6, 5)
Total combinations: 2
- Sum of 12:
- (6, 6)
Total combinations: 1
Now, adding up all the favorable outcomes:
4 (for sum 9) + 3 (for sum 10) + 2 (for sum 11) + 1 (for sum 12) = 10 favorable outcomes.
Step 3: Calculating the Probability
The probability of an event is calculated using the formula:
Probability = (Number of Favorable Outcomes) / (Total Possible Outcomes)
In our case, the probability is:
P(sum ≥ 9) = 10 favorable outcomes / 36 total outcomes
P(sum ≥ 9) = 10/36 = 5/18
Conclusion
Therefore, the probability that the sum of the numbers rolled on two six-sided dice is 9 or higher is:
5/18 or approximately 0.2778 (27.78%).