Question

Please Explain!

Roll a fair die five times. What is the probability of seeing a full house, in the sense that 3 rolls of one type, and two rolls of another different type? Note that we do not allow the 5 rolls to be of the same type.

Answer 1

There are 6 different types for a die - 1,2,3,4,5,6.

The one type of the 3 rolls can be chosen in 6 ways and the other type of the 2 rolls can be chosen in 5 ways. Thus we can choose the values in 6 * 5 = 30 ways.

Also, the 3 rolls of the same type can happen in 5C3 = 10 ways. The remaining two rolls are of the other type.

Thus the full house can happen in 30 * 10 = 300 ways.

In all, since each die roll has 6 types, there are 6⁵​ different possibilities.

The probability is 300/6⁵ = 300/7776 = 0.0386.