We are given a physics class of 40 students. 13 of these students are physics majors. 14 of the students are female. 3 of the physics majors are female. We are asked to find the probability that a randomly chosen student is female or a physics student.
One of the axioms of probability is the additive axiom; the probability of two mutually exclusive events is the sum of the probabilities of the two events or P(A or B)=P(A)+P(B). The key here is that the events must be mutually exclusive.
For two events to be mutually exclusive, the events can never occur at the same time. Alternatively, P(A and B)=0.
Here, being female and a physics student can occur, so the events are not mutually exclusive; thus, we have to use the modified formula:
P(A or B)=P(A)+P(B)-P(A and B).
So, P(female or physics student)=P(female)+P(physics student)-P(female physics student) or `14/40+13/40-3/40=24/40=3/5=.6`
We could approach this from a definition of probability standpoint. Take the size of the event space (the total number of people who are either female or a physics student) divided by the sample space (the total number of people.)
Here the event space has 24 people in it; 14 females plus 13 physics students is 27 people. But we counted the 3 female physics students twice. So, we have 27-3=24 students in the event space, and the probability is 24/40, as before.