If events A and B are mutually exclusive (disjoint), then

P(A or B) = P(A) + P(B)

Otherwise,

P(A or B) = P(A) + P(B) – P(A and B)

In a group of 101 students 30 are freshmen and 41 are sophomores. Find the probability that a student picked from this group at random is either a freshman or sophomore.

Note that P(freshman) = 30/101 and P(sophomore) = 41/101. Thus

P(freshman or sophomore) = 30/101 + 41/101 = 71/101

This makes sense since 71 of the 101 students are freshmen or sophomores.

In a group of 101 students 40 are juniors, 50 are female, and 22 are female juniors. Find the probability that a student picked from this group at random is either a junior or female.

Note that P(junior) = 40/101 and P(female) = 50/101, and P(junior and female) = 22/101. Thus

P(junior or female) = 40/101 + 50/101 – 22/101 = 68/101

This makes sense since 68 of the 101 students are juniors or female.

Not sure why? When we add 40 juniors to 50 females and get a total of 90, we have overcounted. The 22 female juniors were counted twice; 90 minus 22 makes 68 students who are juniors or female.