Mutually Exclusive Events and the Addition Rule

Let B stand for Joe getting hit by one or more buses, C for Joe getting hit by one or more cars, M for Joe getting hit by a motorcycle, T for Joe getting hit by one ore more different kinds of vehicle, S for Joe getting hit by several different kinds of vehicles, and L for Joe getting hit by no vehicles. Are these mutually exclusive events?

You throw a coin 5 times. Choose three events that are mutually exclusive and label them.

If two events are not mutually exclusive then they are overlapping events. For example, assume that at a high school with 300 you count the number of students who speak French and Spanish and make this Venn diagram. How many students speak French? What is the probability that a student speaks French? Let F represent the event that a randomly chosen student speaks French.

Let S represent the event that a randomly chosen student speaks Spanish. Find P(S)

Find the probability that a student speaks both Spanish and French.

Find P(F or S) = P(F and ) + P(F and S) + P( and S) This comes easily from the Venn Diagram.

Find P(F or S) = P(F) + P(S) - P(F and S). We subtract the trilingual speakers because otherwise we would be double counting them in this overlapping event. This comes more easily from a "Klapheck" diagram.

Why is the addition rule P(A or B) = P(A) + P(B) - P(A and B) still true for mutually exclusive events A and B?