Five standard six-sided dice are rolled. We are told there is no three-of-a-kind, but there is a pair of dice that show the same number. These two dice are

4/9

NinjaAnswer: Can you please explain your answer?

To explain NinjaAnswer's answer:

Of the three dice rolled, one may match the original pair and two are different, and this can occur three ways:  ( 1/6 x 5/6 x 5/6 ) + ( 5/6 x 1/6 x 5/6 ) + ( 5/6 x 5/6 x 1/6 )  =  75/216

[ where 1/6 is the probability of matching the first two and 5/6 is the probability of not matching the first two ]

Or, two may match the original pair and one is different:

                     ( 1/6 x 1/6 x 5/6 ) + ( 1/6 x 5/6 + 1/6 ) + ( 5/6 x 1/6 x 1/6 )  =  15/216

Or, all three may match the original pair:     ( 1/6 x 1/6 x 1/6 )  =  1/216

Or, all three may  be the same, but not match the original pair:

                      ( 5/6 x 1/6 x 1/6 )  =  5/216

[ where 5/6 is the probability of getting a new number and 1/6 is the probability of matching that new number ]

Adding these together:  75/216 + 15/216 + 1/216 + 5/216  =  96/216  =  4/9 

Thanks geno3141:

This young person claims that he/she is 12 years old. No matter how precocious, I simply cannot believe that he/she is 12 years old!!.