Four distinct fair of dice are rolled. What is the probability that the sum of the resulting numbers is at least 23?
24 is all sixes, so
P(24) = (1/6)^4
23 is 3 sixes and a five, so there are 4 ways of doing this, with
P(23) = 4*(1/6)^3 * (1/6)
...
To find the probability of the sum of the resulting numbers being at least 23 when four distinct fair dice are rolled, we need to consider all the possible outcomes and count the favorable outcomes.
First, let's find the total number of possible outcomes. Since each dice has six possible outcomes (numbers 1 to 6), the total number of outcomes for all four dice is 6^4 = 1296.
Next, we need to find the number of favorable outcomes where the sum of the resulting numbers is at least 23. We can break this down into different cases:
Case 1: One dice shows a 6, and the sum of the other three dice is at least 17.
There are 4 ways to choose which dice shows a 6, and we can determine the sum of the other three dice by using the formula for the number of ways to choose k items from a set of n items (nCr). In this case, n = 3 (remaining dice) and k = 17 (minimum sum - 6 = 11).
So, the number of outcomes in this case = 3C17 = 1365.
Case 2: One dice shows a 5, and the sum of the other three dice is at least 18.
Similarly, the number of outcomes in this case = 3C18 = 680.
Case 3: One dice shows a 4, and the sum of the other three dice is at least 19.
The number of outcomes in this case = 3C19 = 285.
Case 4: One dice shows a 3, and the sum of the other three dice is at least 20.
The number of outcomes in this case = 3C20 = 60.
Case 5: One dice shows a 2, and the sum of the other three dice is at least 21.
The number of outcomes in this case = 3C21 = 10.
Case 6: One dice shows a 1, and the sum of the other three dice is at least 22.
The number of outcomes in this case = 3C22 = 1.
Now, we can sum up the favorable outcomes from each case to get the total number of favorable outcomes = 1365 + 680 + 285 + 60 + 10 + 1 = 2401.
Finally, we can calculate 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
= 2401 / 1296
≈ 1.85
Therefore, the probability that the sum of the resulting numbers is at least 23 when four distinct fair dice are rolled is approximately 1.85.