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3.3 The Addition Rule

Statistics Notes. 3.3 The Addition Rule. Mutually Exclusive Events. In the previous section, you learned how to find the probability of two events, A and B, occurring in sequence. Such probabilities are denoted by P(A and B). (ex – flipping a heads and rolling a 3)

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3.3 The Addition Rule

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  1. Statistics Notes 3.3 The Addition Rule

  2. Mutually Exclusive Events • In the previous section, you learned how to find the probability of two events, A and B, occurring in sequence. Such probabilities are denoted by P(A and B). (ex – flipping a heads and rolling a 3) • In this section, you will learn how to find the probability that at least one of two events will occur. Probabilities such as these are denoted by P(A or B) and depend on whether the events are mutually exclusive.

  3. Mutually Exclusive Events • Two events A and B are mutually exclusive if A and B cannot occur at the same time. • The Venn diagrams show the relationship between events that are mutually exclusive and events that are not mutually exclusive.

  4. Decide if the events are mutually exclusive. • Event A: Roll a 3 on a die. Event B: Roll a 4 on a die. • Event A: Randomly select a male student.E vent B: Randomly select a nursing major. • Event A: Randomly select a blood donor with type “0” blood.Event B: Randomly select a female blood donor.

  5. Decide if the events are mutually exclusive. • Event A: Randomly select a jack from a standard deck of cardsEvent B: Randomly select a face card from a standard deck of cards • Event A: Randomly select a 20 -year.- old student.Event B: Randomly select a student with blue eyes. • Event A: Randomly select a vehicle that is a Ford.Event B. Randomly select a vehicle that is a Toyota.

  6. The Addition Rule 1 & 2 • Addition Rule 1 • If events A and B are mutually exclusive, then the rule can be simplified to • P(A or B) = P(A) + P(B) • This simplified rule can be extended to any number of mutually exclusive events. • Addition Rule 2 • The probability that events A or B will occur P(A or B) is given by • P(A or B) = P(A) + P(B) - P(A and B).

  7. Examples • A box contains 3 glazed doughnuts, 4 jelly doughnuts, and 5 chocolate doughnuts. If a person selects a doughnut at random, find the probability that it is either a glazed doughnut or a chocolate doughnut. • P(glazed or chocolate)= P(glazed) + P (Chocolate)= 3/12 + 5/12 = 8/12 = 2/3 • The events are Mutually Exclusive.

  8. Examples • At a political rally, there are 20 Republicans, 13 Democrats, and 6 independents. If a person is selected at random, find the probability that he or she is either Democrat or Republican. • P(Democrat or Independent)= P(Democrat) + P(Independent) • =13/39 + 6/39 =19/39

  9. Examples • A day of the week is selected at random. Find the probability that it is a weekend day. • P(Saturday or Sunday) = P(Saturday) + P(Sunday) • P(Saturday or Sunday) = 1/7 + 1/7 = 2/7

  10. Examples • A single card is drawn at random from an ordinary deck of cards. Find the probability that it is either an ace or a black card. • P(ace or black card) = P(ace) + P(black card) • P(ace or black card) = 4/52 + 26/32 – 2/52 = 28/52 = 7/13

  11. Examples • In a hospital units there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are female. If a staff person is selected, find the probability that the subject is a nurse or a male. • P(nurse or male) = P(nurse) + P(male) • P(nurse or male) = 8/13 + 3/13 – 1/13= 10/13

  12. Examples • On New Year’s Eve, the probability of a person driving while intoxicated is 0.32 the probability of a person having a driving accident is 0.09, and the probability of a person having a driving accident while intoxicated is 0.06. What is the probability of a person driving while intoxicated or having a driving accident? • P(intoxicated or accident) = P(intoxicated) + P ( accident) – P(intoxicated and accident) • =.32 + 0.09 – 0.06= 0.35

  13. Finding Probabilities of Mutually Exclusive events • The frequency distribution shows the volume of sales (in dollars) and the number of months a sales representative reached each sales level during the past three years. If this sales pattern continues, what is the probability that the sales representative will sell between $75,000 and $124,999 next month?

  14. Finding Probabilities of Mutually Exclusive events • Find the probability that the sales representative will sell between $0 and $49,999.

  15. Using the Addition Rule to find the Probabilities • A blood bank catalogs the types of blood, including positive or negative Rh-factor, given by donors during the last five days. The number of donors who gave each blood type is shown in the table. A donor is selected at random. • Find the probability that the donor has type 0 or type A blood. • Find the probability that the donor has type B blood or is Rh-negative.

  16. Using the Addition Rule to find the Probabilities • Find the probability that the donor has type B or type AB blood. • Find the probability that the donor has type O blood or is Rh-positive.

  17. A summary of Probability!

  18. Combining Rules to Find Probability • Use the graph at the right to find the probability that a randomly selected draft pick is not a running back or a wide receiver.

  19. Combining Rules to Find Probability • Find the probability that a randomly selected draft Pick is not a linebacker or a quarterback.

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