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Solution: To calculate the probability that the sum of four 6-sided dice equals 14, we begin by determining the total number of possible outcomes:
Next, we count the number of integer solutions to the equation:
We now count the number of non-negative integer solutions to $a + b + c + d = 10$ with each variable at most 5.
We use the principle of inclusion-exclusion. First, count the total number of non-negative integer solutions without restrictions:
Now subtract the number of solutions where at least one variable exceeds 5. Lets define $A_i$ as the set where variable $i$ is at least 6.
So no such cases. Therefore, total valid solutions:
Question: An ichthyologist is studying a population of fish in a lake and tags 10 fish. Later, a sample of 15 fish is randomly caught. What is the probability that exactly 4 of the tagged fish are in the sample, assuming the lake contains 200 fish total?
Solution: This is a classic hypergeometric probability problem. We are selecting 15 fish from a population of 200, of which 10 are tagged. We want the probability that exactly 4 of the 15 sampled fish are tagged.
The hypergeometric probability formula is:
$k = 4$ (desired tagged fish in sample).