So say there's 19 bikes that were returned to a store because of 3 different reasons: 4 were defective, 6 were not the correct brand, and the rest were not the right color.

Now there's three randomly selected customer files (for an audit), and these files can be selected more than once (i.e. this experiment is done with 'replacement').

How would I go about finding the probability that in all three cases, the bikes were returned for the same reason?

I'm trying to picture this with three separate cases of intersecting Venn diagrams, but after this I'm unsure of how to proceed since there's the possibility of replacement.

Thanks for the help


With replacement, the probability on each draw that a bike is defective is $p_1=4/19$, the probability that it's not the correct brand $p_2=6/19$, and $p_3=10/19$ for wrong color. The chance all three files are the same is then $p_1^3+p_2^3+p_3^3$.


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