A box contains five green balls, three black balls, and seven red balls. Two balls are selected at random without replacement from the box. What it the probability that both balls are the same color?

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We will denote the probability of randomly selecting two green balls as P(G).  The probability of selecting a green ball first is 5/15 (or 1/3).  Without replacing the first green ball, the probability of selecting a green ball on the second pick is 4/14 (or 2/7).  Thus the probability of randomly selecting two green balls is (1/3)(2/7) = 2/21.Likewise, the probability of randomly selecting two black balls is P(B) = (3/15)(2/14) = 1/35.And once more, the probability of randomly selecting two red balls is P(R) = (7/15)(6/14) = 1/5.These three options are mutually exclusive since they cannot occur at the same time.  Therefore, the probability that G, B, or R will occur is the sum of the probability of each event.  That is P(G or  B or R) = P(G) + P(B) + P(R) = 2/21 + 1/35 + 1/5 = 34/105.