Two different integers are randomly chosen from the set $$\{ -5, -8, 7, 4, -2 \}.$$ What is the probability that their product is negative? Express your answer as a common fraction.
To find the probability, I need to count the number of favorable outcomes and divide by the total number of possible outcomes. The product of two integers is negative if and only if one of them is positive and the other is negative. In the given set, there are two positive integers (7 and 4) and three negative integers (-5, -8, and -2). Therefore, the number of ways to choose one positive and one negative integer is $$2 \times 3 = 6.$$ The total number of ways to choose two different integers from the set is $$\binom{5}{2} = \frac{5 \times 4}{2} = 10.$$ The probability is then $$\frac{6}{10} = \frac{3}{5}.$$ 