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Probability of events that are not mutually exclusive

If two events, A and B, are not mutually exclusive, then the probabilitythat either A or B occurs is the sum of their probabilities decreased by the probability of both occurring.

P(A or B)=p(A)+p(B)-p(A and B)

Of 30 boys, 12 are on the Honor Roll, 10 play sports, and 15 are on the Honor Roll and play sport. What is the probability that a randomly selected student plays sports or is on the Honor Roll?

Since some students play varsity sports and are on the Honor Roll, the events are
not mutually exclusive
.

$p(sports)=\frac{10}{30}$

$p(Honor\;Roll)=\frac{12}{30}$

$p(sports\;and\;Honor\;Roll)=\frac{15}{30}$

 p(sport or Honor Roll) = p(sport)+p(Honor Roll)-p(sport and Honor Roll) = $\frac{10}{30}+\frac{12}{30}-\frac{15}{30}$ = $\frac{7}{30}$