When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to four decimal places.

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1 Chapter 13: General Rules of Probability Notation for Probability Event: an outcome that is usually denoted by a capital letter; A, B, C. A, B, C: Specific events P: A probability P(A): The probability of event A occurring (1) The probability of an impossible event is 0. (2) The probability of an event that is certain to occur is 1. (3) 0 P (A) 1 (4) Rounding Off Probabilities When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to four decimal places. Addition Rule (or): Either one or the other or both. Addition Rule (1) Disjoint case: P (A B) = P (A) + P (B) P (A or B) = P (A) + P (B) At a political rally, there are 20 Republicans, 13 Democrats, and 16 Independents. If a person is selected, find the probability that he or she is either a Democrat or an Independent. Addition Rule (2) Not Disjoint case: P (A B) = P (A) + P (B) P (A B) P (A or B) = P (A) + P (B) P (A and B) In hospital unit there are eight nurses and five physicians. Seven nurses and three physicians are females. One nurse and two physicians are males. If a staff person is selected, find the probability that the subject is nurse or a male.

2 Staff Females Males Total Nurses Physicians Total For a card drawn from an ordinary deck, find the probability of getting a queen or a king. For a card drawn from an ordinary deck, find the probability of getting a queen or a club. Complement Events: The complement of event A consists of all outcomes in which event A does not occur, denoted by A. P(A): The probability of the complement of event A P (A) + P (A) = 1 P (A) = 1 P (A) P (A) = 1 P (A) If two dice are rolling, find the probability getting 5. If two dice are rolling, find the probability not getting 5.

3 Independence and the multiplication rule Multiplication Rule (and) Multiplication Rule (1) When two events are independent, the probability of both occurring is P (A and B) = P (A) P (B) What does Independence mean? Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. 1. A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and 4 on the die. 2. A card is drawn from a deck and replaced; then a second card is drawn. Find the probability of getting a queen both times.

4 Multiplication Rule (2) When two events are dependent, the probability of both occurring is P (A and B) = P (event A occurs in a first trial and event B occurs in a second trial) P (A and B) = P (A) P (B A) 3. A card is drawn from a deck and if the card will not be replaced; then a second card is drawn. Find the probability of getting a queen both times. 4. In a shipment of 25 microwave ovens, 2 are defective. If two ovens are randomly selected and tested, find the probability that both are defective if the first one is not or (without) replaced after it has been tested. 5. Three cards are drawn from an ordinary deck and not or (without) replaced. Find the probability of the following. 1) Getting three jacks. 2) Getting an ace, a king, and a queen in order.

5 Conditional probability: Find the probability of an event when we have additional information that some other event has already occurred. P (B A) = P (A and B) P (A) This is called the conditional probability of B given A. 1. A recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results of the survey are shown in the table. Gender Yes No Male Female 8 42 a) The respondent answered yes given that the respondent was a female. b) The respondent was a male given that the respondent answered no. 2. Eighty students in a school cafeteria were asked if they favored a ban on smoking in the cafeteria. The results of the survey are shown in the table. Class Favor Oppose No opinion - Freshman Sophomore a) The student opposes the ban given that he or she is a freshman. b) The student is a sophomore given that the student favors the ban.

6 Complements (As a tool): Probability of at least one : Find the probability that among several trials, we get at least one of some specified event. At least one is equivalent to one or more. P (at least one) = 1 P (none) 1. A coin is tossed four times. Find probability of getting at least one tail. 2. A game is played by drawing four cards from an ordinary deck and replacing each card after it is drawn. find the probability of winning if at least one ace is drawn. 3. A student answered five multiple choice questions. Each question had four possible choices for its answer. Find the probability of answering at least one question with the correct answer.