The probability that after re-rolling these three dice, at least three of the five dice show the same value is
Of the three dice rolled, one may match the original pair and two are different, and this can occur three ways:
[ where is the probability of matching the first two and
is the probability of not matching the first two ]
Or, two may match the original pair and one is different
Or, all three may match the original pair
Or, all three may be the same, but not match the original pair:
[ where is the probability of getting a new number and
is the probability of matching that new number ]
Adding these together:
To learn more about probability with the given link
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