Suppose that in a state lottery game players choose three digits, each from the set 0 to 9, as they do in many state lottery games. But for this game, what counts is the sum of the three chosen digits. The state selects a winning sum from the possibilities 0 (0 + 0 + 0) to 27 (9 + 9 + 9) by randomly choosing three digits and using their sum. For instance, if the three digits 483 are selected, then a winning ticket is any ticket in which the three digits sum to 4 + 8 + 3 =15. As an example, a ticket with 195 would win because 1 + 9 + 5 = 15.
a. Using the physical assumption that all 1000 choices (000, 001, ... , 999) are equally likely, find the probability that three digits chosen at random will sum to 15, so that the winning sum is 15.
b. Simulate playing this game 1000 times, randomly selecting the three digits each time. Based on the simulation, find the approximate probability that the winning sum is 15.
c. Based on information about how people respond when asked to pick random digits, do you think the proportion of people who would actually pick three digits summing to 15 is close to the probabilities you found in parts (a) and (b)? Explain.