The Addition Rule and the Rule of Complements Section 5.2

Objectives Compute probabilities by using the General Addition Rule Compute probabilities by using the Addition Rule for Mutually Exclusive Events Compute probabilities by using the Rule of Complements

Compute probabilities by using the General Addition Rule Objective 1 Compute probabilities by using the General Addition Rule

A or B Events and the General Addition Rule A compound event is an event that is formed by combining two or more events. One type of compound event is of the form A or B. The event A or B occurs whenever A occurs, B occurs, of A and B both occur. Probabilities of events in the form A or B are computed using the General Addition Rule. For any two events A and B, 𝑃(A or B) = 𝑃(A) + 𝑃(B) – 𝑃(A and B) The General Addition Rule

Example – The General Addition Rule 1000 adults were asked whether they favored a law that would provide support for higher education. In addition, each person was classified as likely to vote or not likely to vote based on whether they voted in the last election. What is the probability that a randomly selected adult is likely to vote or favors the law? Favor Oppose Undecided Likely to vote 372 262 87 Not likely to vote 151 103 25 Solution: There are 372 + 262 + 87 = 721 people who are likely to vote, so 𝑃(Likely to vote) = 721/1000 = 0.721. There are 372 + 151 = 523 people who favor the law, so 𝑃(Favor) = 523/1000 = 0.523. The number of people who are both likely to vote and who favor the law is 372. Therefore, 𝑃(Likely to vote AND Favors) = 372/1000 = 0.372. By the General Addition Rule, 𝑃(Likely to vote or Favors) = 𝑃(Likely to vote) + 𝑃(Favors) – 𝑃(Likely to vote and favors) = 0.721 + 0.523 – 0.372 = 0.872

Objective 2 Compute probabilities by using the Addition Rule for Mutually Exclusive Events

Mutually Exclusive Events Two events are said to be mutually exclusive if it is impossible for both events to occur. Example: A die is rolled. Event A is that the die comes up 3, and event B is that the die comes up an even number. These events are mutually exclusive since the die cannot both come up 3 and come up an even number. A fair coin is tossed twice. Event A is that one of the tosses is heads, and Event B is that one of the tosses is tails. These events are not mutually exclusive since, if the two tosses are HT or TH, then both events occur.

The Addition Rule for Mutually Exclusive Events If events A and B are mutually exclusive, then 𝑃(A and B) = 0. This leads to a simplification of the General Addition Rule. If A and B are mutually exclusive events, then 𝑃(A or B) = 𝑃(A) + 𝑃(B) Addition Rule for Mutually Exclusive Events

Example – Addition Rule/Mutually Exclusive In the 2012 Olympic Games, a total of 10,735 athletes participated. Of these, 530 represented the United States, 277 represented Canada, and 102 represented Mexico. What is the probability that an Olympic athlete chosen at random represents the U.S. or Canada? Solution: These events are mutually exclusive, because it is impossible to compete for both the U.S. and Canada. So, 𝑃(U.S. or Canada) = 𝑃(U.S.) + 𝑃(Canada) = 530 10,735 + 277 10,735 = 807 10,735 =0.07517

Compute probabilities by using the Rule of Complements Objective 3 Compute probabilities by using the Rule of Complements

Complements If there is a 60% chance of rain today, then there is a 40% chance that it will not rain. The events “Rain” and “No rain” are complements. The complement of an event says that the event does not occur. Example: Two hundred students were enrolled in a Statistics class. Find the complements of the following events: Exactly 50 of them are business majors. The complement is that the number of business majors is not 50. More than 50 of them are business majors. The complement is that 50 or fewer are business majors. At least 50 of them are business majors. The complement is that fewer than 50 are business majors. If A is any event, the complement of A is the event that A does not occur. The complement of A is denoted Ac.

The Rule of Complements Example: According to the Wall Street Journal, 40% of cars sold in July 2013 were small cars. What is the probability that a randomly chosen car sold in July 2013 is not a small car? Solution: 𝑃(Not a small car) = 1 – 𝑃(Small car) = 1 – 0.40 = 0.60. 𝑃(Ac) = 1 – 𝑃(A) The Rule of Complements

You Should Know… How to use The General Addition Rule to compute probabilities of events in the form A or B How to determine whether events are mutually exclusive How to compute probabilities of mutually exclusive events How to determine the complement of an event How to use the Rule of Complements to compute probabilities