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?
Observing the outcome of a very large number N of trials, we pick out all the trial that include . The number of those chosen trials will be : N*[1-0.75^7].
We now remove only one Y hat from each of the chosen outcomes, which leaves us with the above found number of outcomes which have now only 6 hats in each group.
The number of cases (out of the chosen group) which include at least one B hat, will be : N*[1-0.75^7]*[1-0.75^6] .
Out of this new group we remove now one B hat from each outcome, and continue in the same fashion with the remaining two colors, bringing us to a remaining group of outcomes that originally contained at least one of each color. So the probability of having all 4 colors in a trial will be :
[1-0.75^7]*[1-0.75^6]*[1-0.75^5]*[1-0.75^4] = 0.37137