Mutually Exclusive or Not In order to develop a sample space for a probability problem, you have to accurately count the items there.... First case: Two sets A and B have no elements in common, they are called ____________________________ Second case: If two events, A and B, are ________________________, then they _________. disjoint or mutually exclusive. not mutually exclusive overlap
Notation: Is the intersection of sets A and B. It represents the elements that both sets have in common. Is the union of sets A and B. It represents all of the elements that are in both sets.
Addition Counting Principle: (Mutually Exclusive Form) If two finite sets A and B are mutually exclusive, then
Probability A statement on probability can come from this – divide both sides by the number of items in the sample space, and you get a theorem on probability for mutually exclusive events:
Example 1: Toss a fair die. What is the probability that a 2 or a 3 is tossed? How many in the sample space? _____ Set A = tossing a 2 outcomes = _____ Set B = tossing a 3 outcomes = _____ Therefore,
An Extra Example: Toss two fair die. What is the probability the sum of the two die is 2 or 3? How many in the sample space? _____ Set A = tossing a sum of 2 outcomes = _____ Set B = tossing a sum of 3 outcomes = _____ Therefore,
Addition Counting Principle (General Form) (formula can be used for all sets – mutually exclusive or not mutually exclusive) For any finite sets A and B, Like before, dividing by the number in the sample space yields:
If A and B are any events in the same finite sample space, then Theorem: Probability of a Union of Events – General Form
Example 2: Participants at a two-day conference could register for only one of the days or both. There were 231 participants on Friday and 252 on Saturday. The total number of people who registered for the conference was 350. Illustrate this situation. a. How many people attended both days? 231252
b.Suppose the name of a participant is drawn for a door prize. What is the probability that this person attended on both days? (show 2 methods to find answer) 1 st use definition of probability Example 2:
2 nd method: use general form of the probability of a union of events
Complementary Events: The complement of set A is called ________. Theorem: Probability of Complements: If A is any event, then not A
Example 3: Two dice are tossed. Find the probability of each event. a.Their sum is seven. b.Their sum is not seven.
Example 4: If the probability of giving birth to a boy is ¼, what is the probability of not giving birth to a boy?
Example 5: A bag of M&M’s contains 12 red, 11 yellow, 5 green, 6 orange, 5 blue, and 16 brown candies. a. What is the probability that you randomly choose a yellow M&M? b. What is the probability that if you randomly choose another candy, it is not a green M&M?
Closure If two sets are mutually exclusive, what do they have in common? What is the general form of the Probability of the Union of Two Sets? What is the complement of set A? Suppose two fair dice are rolled. Event A is the first die is 6. Event B is the sum of the dice is 10. – Are the two sets mutually exclusive? – Find N(A B) – Find N( A B)