Seven sportsmen (named A,B,C,D,E,F and G) buy 7 hats choosing randomly out of 4 available
colors (say G,R,W and Y).

What is the probability that all
the colors were chosen?

Each sportsmen has a 3/4 chance of excluding a given color. Starting with the color Y,there is a .133 (3/4 ^7) chance that none of the sportsmen will pick it. But there are 4 possible colors which may be excluded by all 7 sportsmen. So multiply .133 * 4 = .5339. That is probability that any single color will be excluded by all 7 sportsmen. So the probability that this won't happen (and all colors will be chosen) is 1 - .5339, or

.4661