If four standard six-sided dice are rolled,what is the probability that the sum of the number on the top faces is 22, or 23,or 24,.express your answer as a common fraction
To find the probability of rolling a sum of 22, 23, or 24 with four standard six-sided dice, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's calculate the total number of possible outcomes when rolling four dice. Since each die has 6 sides, we can obtain the number of possible outcomes by raising 6 to the power of 4 (6^4) since there are four dice. This gives us a total of 1296 possible outcomes.
Next, we need to determine the number of favorable outcomes, i.e., the number of outcomes that sum up to 22, 23, or 24. We can approach this by using a systematic counting method.
Case 1: Sum of 22
To roll a sum of 22, we can consider each dice individually.
If the first die shows 6, we need the remaining three dice to sum up to 16 (22 - 6).
If the first die shows 5, we need the remaining three dice to sum up to 17 (22 - 5).
If the first die shows 4, we need the remaining three dice to sum up to 18 (22 - 4).
...
If the first die shows 1, we need the remaining three dice to sum up to 21 (22 - 1).
Observing this pattern, we see that we need to find the number of solutions to the equation x1 + x2 + x3 = 16, where each xi represents the number on the remaining dice (xi can only be between 1 and 6, inclusive). This is a classic "stars and bars" problem and can be solved using combinatorics. The number of solutions is C(15, 2), which equals 105.
We need to do this separately for each of the cases (sums of 16, 17, 18, ..., 21). Adding up these counts for each case, we get the total number of favorable outcomes for a sum of 22.
Case 2: Sum of 23
Using the same logic as above, we need to find the number of solutions to the equation x1 + x2 + x3 = 17. Applying the "stars and bars" formula, we get C(16, 2) = 120 as the number of favorable outcomes for a sum of 23.
Case 3: Sum of 24
Similarly, we need to find the number of solutions to the equation x1 + x2 + x3 = 18, which gives us C(17, 2) = 136 favorable outcomes.
Finally, we add up the counts for all three cases to get the total number of favorable outcomes.
Total favorable outcomes = 105 + 120 + 136 = 361
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Favorable outcomes / Total outcomes
Probability = 361 / 1296 = 361/1296
Therefore, the probability of obtaining a sum of 22, 23, or 24 when rolling four standard six-sided dice is 361/1296.