Two regular 6 sided dice are tossed. Compute the probability that the sum of the pips on the upward faces of the two dice is the following. At least 8
To compute the probability that the sum of the pips on the two dice is at least 8, we need to find the number of favorable outcomes and divide it by the total number of possible outcomes.
Let's first determine the favorable outcomes. The possible sums that are at least 8 are 8, 9, 10, 11, and 12. We can calculate the number of ways to obtain each sum using a table:
| Dice 1 | Dice 2 | Sum |
|--------|--------|-----|
| 2 | 6 | 8 |
| 3 | 5 | 8 |
| 4 | 4 | 8 |
| 5 | 3 | 8 |
| 6 | 2 | 8 |
| 3 | 6 | 9 |
| 4 | 5 | 9 |
| 5 | 4 | 9 |
| 6 | 3 | 9 |
| 4 | 6 | 10 |
| 5 | 5 | 10 |
| 6 | 4 | 10 |
| 5 | 6 | 11 |
| 6 | 5 | 11 |
| 6 | 6 | 12 |
There are 15 favorable outcomes.
Now, let's calculate the total number of possible outcomes. Since each dice has 6 sides, there are 6 possible outcomes for each dice. Therefore, the total number of possible outcomes is 6 * 6 = 36.
Finally, we can compute the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 15 / 36
Probability = 0.4167 (rounded to four decimal places)
Therefore, the probability that the sum of the pips on the upward faces of the two dice is at least 8 is approximately 0.4167.
To compute the probability that the sum of the pips on the upward faces of the two dice is at least 8, we need to find the number of favorable outcomes and the total number of possible outcomes.
Let's break it down into cases:
Case 1: The sum is 8.
We can obtain a sum of 8 in three ways: (2, 6), (3, 5), or (4, 4). So, there are 3 favorable outcomes.
Case 2: The sum is 9.
We can obtain a sum of 9 in four ways: (3, 6), (4, 5), (5, 4), or (6, 3). So, there are 4 favorable outcomes.
Case 3: The sum is 10.
We can obtain a sum of 10 in three ways: (4, 6), (5, 5), or (6, 4). So, there are 3 favorable outcomes.
Case 4: The sum is 11.
We can obtain a sum of 11 in two ways: (5, 6) or (6, 5). So, there are 2 favorable outcomes.
Case 5: The sum is 12.
We can obtain a sum of 12 in one way: (6, 6). So, there is 1 favorable outcome.
Therefore, the total number of favorable outcomes is 3 + 4 + 3 + 2 + 1 = 13.
The total number of possible outcomes when two dice are tossed is 6 × 6 = 36 (as each die has 6 sides).
Hence, the probability of obtaining a sum of at least 8 is given by 13/36 (13 favorable outcomes out of 36 possible outcomes), which can be simplified as approximately 0.3611 or 36.11%.
There are 36 possible outcomes when rolling two dice, as each die has 6 possible outcomes. To find the probability that the sum of the pips is at least 8, we need to count how many of these outcomes result in a sum that is at least 8.
One way to approach this is to list all of the possible outcomes and count how many of them have a sum of at least 8:
- (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
- (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
- (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
- (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
- (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
There are 25 outcomes in total, and 15 of them have a sum of at least 8. Therefore, the probability of getting a sum of at least 8 is:
15/36 = 5/12
So the probability of rolling two dice and getting a sum of at least 8 is 5/12.