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Addition Rule in Probability

Learn how to use the addition rule to find the probability of two events, whether they are mutually exclusive or non-mutually exclusive. Understand the concept of inclusive "or" and avoid double counting probabilities. Examples and explanations provided.

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Addition Rule in Probability

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  1. Section 3.3 The Addition Rule

  2. War Warm Up • Nick picks marbles from a jar that contains 3 red, 2 blue, and 5 green marbles. What is the probability that Nick picks a green marble given that it was not blue? • Jamie picks two cards from a standard deck of cards (without replacement). What is the probability that Jamie chooses a queen on her second pick given that she chose a queen on her first pick?

  3. How to determine if two events are mutually exclusive How to use the addition rule to find the probability of two events. Objectives/Assignment

  4. In probability and statistics, the word “or” is usually used as an “inclusive or” rather than an “exclusive or.” For instance, there are three ways for “Event A or B” to occur. A occurs and B does not occur B occurs and A does not occur A and B both occur What is different?

  5. Students often confuse the concept of independent events with the concept of mutually exclusive events. Independent does not mean mutually exclusive

  6. By subtracting P(A and B), you avoid double counting the probability of outcomes that occur in both A and B. Study Tip

  7. Compare “A and B” to “A or B” B B A A The compound event “A and B” means that A and B both occur in the same trial. Use the multiplication rule to find P(A and B). The compound event “A or B” means either A can occur without B, B can occur without A or both A and B can occur. Use the addition rule to find P(A or B). A or B A and B

  8. Mutually Exclusive Events Two events, A and B, are mutually exclusive if they cannot occur in the same trial. A = A person is under 21 years old B = A person is running for the U.S. Senate A = A person was born in Philadelphia B = A person was born in Houston A Mutually exclusive B P(A and B) = 0 When event A occurs it excludes event B in the same trial.

  9. Non-Mutually Exclusive Events B A Non-mutually exclusive P(A and B) ≠ 0 If two events can occur in the same trial, they are non-mutually exclusive. A = A person is under 25 years old B = A person is a lawyer A = A person was born in Philadelphia B = A person watches West Wing on TV A and B

  10. The Addition Rule The probability that one or the other of two events will occur is: P(A) + P(B) – P(A and B) A card is drawn from a deck. Find the probability it is a king or it is red. A = the card is a king B = the card is red. P(A) = 4/52 and P(B) = 26/52 butP(A and B) = 2/52 P(A or B) = 4/52 + 26/52– 2/52 = 28/52 = 0.538

  11. The Addition Rule A card is drawn from a deck. Find the probability the card is a king or a 10. A = the card is a king B = the card is a 10. P(A) = 4/52 and P(B) = 4/52 and P(A and B) = 0/52 P(A or B) = 4/52 + 4/52 – 0/52 = 8/52 = 0.054 When events are mutually exclusive, P(A or B) = P(A) + P(B)

  12. Contingency Table The results of responses when a sample of adults in 3 cities was asked if they liked a new juice is: Omaha Seattle Miami Total Yes 100 150 150 400 No 125 130 95 350 Undecided 75 170 5 250 Total 300 450 250 1000 One of the responses is selected at random. Find: 3. P(Miami or Yes) 4. P(Miami or Seattle) 1. P(Miami and Yes) 2. P(Miami and Seattle)

  13. Contingency Table Omaha Seattle Miami Total Yes 100 150 150 400 No 125 130 95 350 Undecided 75 170 5 250 Total 300 450 250 1000 One of the responses is selected at random. Find: 1. P(Miami and Yes) 2. P(Miami and Seattle) = 250/1000 • 150/250 = 150/1000 = 0.15 =0

  14. Contingency Table Omaha Seattle Miami Total Yes 100 150 150 400 No 125 130 95 350 Undecided 75 170 5 250 Total 300 450 250 1000 3 P(Miami or Yes) 4. P(Miami or Seattle) 250/1000 + 400/1000 – 150/1000 = 500/1000 = 0.5 250/1000 + 450/1000 – 0/1000 = 700/1000 = 0.7

  15. Probability at least one of two events occur P(A or B) = P(A) + P(B) - P(A and B) Add the simple probabilities, but to prevent double counting, don’t forget to subtract the probability of both occurring. Summary For complementary events P(E') = 1 - P(E) Subtract the probability of the event from one. The probabilityboth of two events occur P(A and B) = P(A) • P(B|A) Multiply the probability of the first event by the conditional probability the second event occurs, given the first occurred.

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