Two events are mutually exclusive if they can never happen together.[br]If P(A and B)=0, then events A and B are mutually exclusive.
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.[br]Are these mutually exclusive events?[br]
You throw a coin 5 times. Choose three events that are mutually exclusive and label them.
Let A represent getting no heads. B represent getting two or three heads. C represents getting more than three heads.
[img]https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A9e38ae102a37cb33e00387372e33deeea9a01e74cca6ce89b19de92e%2BCOVER_PAGE_TINY%2BCOVER_PAGE_TINY.1[/img][br]The addition rule for mutually exclusive events is easy. To find the probability that either A or B occurs add their individual probabilities together.
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. [img]https://www.mathsisfun.com/data/images/set-language-ex2.svg[/img]How many students speak French? What is the probability that a student speaks French?[br]Let F represent the event that a randomly chosen student speaks French.
9+7 = 16 = |F|[br]P(F) = 16/300 = 0.05333[br]
Let S represent the event that a randomly chosen student speaks Spanish.[br]Find P(S)
7+14 = 21 = |S|[br]P(S) = 21/300 = 0.07
Find the probability that a student speaks both Spanish and French.
P(S and F) = 7/300 = 0.02333333333[br]Notice this is not = 0
Find P(F or S) = P(F and [math]S^C[/math]) + P(F and S) + P([math]F^C[/math] and S)[br]This comes easily from the Venn Diagram.
P(F or S) = 9/300 + 7/300 + 14/300 = 30/300 = 0.1
Find P(F or S) = P(F) + P(S) - P(F and S).[br]We subtract the trilingual speakers because otherwise we would be double counting them in this overlapping event.[br]This comes more easily from a "Klapheck" diagram.
16/300 + 21/300 - 7/300 = 30/300 = 0.1
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?
P(A or B) = P(A) + P(B) - P(A and B)[br] = P(A) + P(B)[br]since P(A and B) = 0 for mutually exclusive events.