Final answer:
Calculating the expected value for the number of distinct colored balls Billy will select involves a probability model that accounts for combinations of draws with replacement. The problem requires considering the probabilities for each outcome and is an exercise in probability and expected value within mathematics.
Step-by-step explanation:
The student's question asks about the expected value for the number of distinct colored balls selected when Billy randomly chooses 4 balls from a box with a yellow, an orange, a green, and a blue ball, with replacement. This is a problem related to probability and expected value, a concept in mathematics that refers to the average outcome one expects to occur after many iterations of an experiment.
The method of calculating the expected number of distinct colors selected is not straightforward because it involves considering the probability of each possible number of distinct balls being drawn. Each draw is independent, and on each draw, there is a ¼ probability of selecting any given color ball since there are 4 colors and the selection is with replacement. However, to calculate the exact expected value, it would require setting up and solving a probability model that accounts for all possible combinations of draws and the corresponding numbers of distinct balls that result from those draws.
A simplified approach to understand this might be to simulate the process with theoretical probabilities or empirically through repeated trials, and analyze how the average number of distinct balls trends over time. However, without the actual calculations or simulations, specifying the expected value is not practical.