What is the probability of getting either a sum of 5 or at least one 5 in the roll of a pair of dice?

In a throw of 2 fair dice, there are 6*6=36 equiprobability outcomes.

To get a sum of 5, there are 4 ways, (1,4),(2,3),(3,2),(4,1) with probability of 4/36=1/9

To get at least one 5, there are 6+6-1=11 outcomes (note (5,5) has been counted in both, so subtracted from sum).  The probability is 11/36

Since the two events are mutually exclusive (once we have a five, the sum can no longer be 5), we can add the probabilities to get the probability of one event or the other.

P(sum of 5 OR at least one 5)=1/9+11/36=4/36+11/36=15/36=5/9

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