7) A small pond contains six catfish and four bluegill. If six fish are caught at random, what is the probability that at most four catfish have been caught?

To approach this problem, we need to calculate the probability of catching at most four catfish when six fish are caught from a pond containing six catfish and four bluegill. This includes the scenarios where 0, 1, 2, 3, or 4 catfish are caught, leaving the rest to be bluegill. We can use combinatorial formulas (binomial coefficients) to evaluate the number of successful outcomes for each case and then divide by the total possible outcomes of catching six fish. The total number of ways to catch any 6 fish from 10 is given by \(\binom{10}{6}\). Each specific scenario will also involve combinations of catfish and bluegill caught, so we can apply the counts accordingly. Once you've calculated all valid cases and summed their probabilities, you'll arrive at the final probability for the scenario. Don’t forget to keep an eye on your calculations to avoid miscounts, as they can easily stack up in combinatorial problems!