8.2 Union, Intersection, and Complement of Events; Odds


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1 8.2 Union, Intersection, and Complement of Events; Odds Since we defined an event as a subset of a sample space it is natural to consider set operations like union, intersection or complement in the context of events. Union and Intersection Definition (Union and Intersection of Events) If A and B are two events in a sample space S, then the union of A and B is an event, denoted by A B, is defined as A B = {e S e A or e B}, and the intersection of A and B is an event, denoted by A B, is defined as A B = {e S e A and e B}. A B is also referred to as event A or B, and A B is referred to as event A and B. Definition (Mutually Exclusive Events) Events A and B are called mutually exclusive if their intersection is the empty set, i.e, A B =.
2 Consider the sample space of equally likely events for the rolling of a single fair die: What is the probability of rolling an odd number and a prime number? What is the probability of rolling an odd number or a prime number? 2
3 Question: If A and B are events in a sample space S, how is the probability of A B related to the individual probabilities of A and of B? Theorem 1 (Probability of the Union of Two Events) For any events A and B, P (A B) = P (A) + P (B) P (A B). (1) Note: If A and B are mutually exclusive, then A B = and (1) becomes P (A B) = P (A) + P (B) P ( ) }{{} =0 = P (A) + P (B). Example 1 Suppose that two fair dice are rolled. (a) What is the probability that a sum of 2 or 3 turns up? 3
4 (b) What is the probability that both dice turn up the same or that a sum greater than 8 turns up? 4
5 Example 2 What is the probability that a number selected at random from the first 140 positive integers is (exactly) divisible by 4 or 6? 5
6 Complement of an Event Definition (Complement of an Event) If E is an event in a sample space S, then the complement of E relative to S, denoted by E, is defined as E = {e S e is not in E (e / E)} Note: E and E are mutually exclusive, and E E = S. Note: E is the complement of E, i.e., (E ) = E. Consider the sample space of equally likely events for the rolling of a single fair die: If E is the event of getting a number that is less than 2, i.e, Then E, the complement of E, is or, equivalently, E is the event of getting a number that is grater or equal than 2. 6
7 Question: How do the probabilities of E and E may be related? Since S = E E, applying Theorem 1, we get S = E E P (S) = P (E E ) 1 = P (E) + P (E ) P (E E ) 1 = P (E) + P (E ) P ( ) }{{} =0 1 = P (E) + P (E ) Theorem 2 (Probability of the Complement of an Event) For any event E, P (E ) = 1 P (E), P (E) = 1 P (E ). Example 3 If two fair dice are rolled what is the probability of getting the sum of the two resulting numbers being less or equal than 10? 7
8 Example 4 A shipment of 40 precision parts, including 8 that are defective, is sent to an assembly plant. The quality control division selects 10 at random for testing and rejects the entire shipment if 1 or more in the sample are found to be defective. What is the probability that the shipment will be rejected? 8
9 Odds When the probability of an event E is known, it is often customary to speak of odds for or against E rather than the probability of E. Definition (Odds of an Event) Given an event E, odds for E = P (E) 1 P (E) = P (E), provided P (E) 1, P (E ) odds against E = P (E ) 1 P (E ) = P (E ), provided P (E) 0. P (E) Example 5 (a) If 3 cards are picked from a standard 52card deck, what is the probability that all 3 cards are spades? (b) If 3 cards are picked from a standard 52card deck, what are the odds that all 3 cards are spades? 9
10 Example 6 If in repeated rolls of two fair dice, the odds against rolling a 6 before rolling a 7 are 6 to 5, then what is the probability of rolling a 6 before rolling a 7? 10
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