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Warm-Up: Define sample space. Give the sample space for the sum of the numbers for a pair of dice.

Warm-Up: Define sample space. Give the sample space for the sum of the numbers for a pair of dice. You flip four coins. What’s the probability of getting exactly two heads? (Hint: List the outcomes first).

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Warm-Up: Define sample space. Give the sample space for the sum of the numbers for a pair of dice.

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  1. Warm-Up: • Define sample space. • Give the sample space for the sum of the numbers for a pair of dice. • You flip four coins. What’s the probability of getting exactly two heads? (Hint: List the outcomes first). • Joey is interested in investigating so-called hot streaks in foul shooting among basketball players. He’s a fan of Carla, who has been making approximately 80% of her free throws. Specifically, Joey wants to use simulation methods to determine Carla’s longest run of baskets on average, for 20 consecutive free throws. a) Describe a correspondence between random digits from a random digit table and outcomes. b) What will constitute one repetition in this simulation? c) Starting with line 101 in the random digit table, carry out 4 repetitions and record the longest run for each repetition. d) What is the mean run length for the 4 repetitions?

  2. P(A U B) = P(A) + P(B) => “A or B” “A union B” is the set of all outcomes that are either in A or in B.

  3. Disjoint/Complement

  4. Probability Distribution • Find the probability that the student is not in the traditional undergraduate age group of 18-23 • Find P(30+ years)

  5. Find P(A), P(B), P(C) Find P(A’), P(B’), P(C’) Venn Diagram

  6. Find: P(AUB) =getting an even number or a number greater than or equal to 5 or both P(AUC) =getting an even number or a number less than or equal to 3 or both P(BUC)=getting a number that is at most 3 or at least 5 or both. Venn Diagram: Union (“Or”/Addition Rule)

  7. Remember this?

  8. Ex. 6.14, p. 420 • Because all 36 outcomes together must have probability 1 (Rule 2), each outcome must have probability 1/36. • Now find P(rolling a 7) =

  9. Consider the events A = {first digit is 1}, B = {first digit is 6 or greater}, and C = {a first digit is odd} • Find P(A) and P(B) • Find P(complement of A) • Find P(A or B) • Find P(C) • Find P(B or C)

  10. Ex. 6.16, p. 422 Find the probability of the event B that a randomly chosen first digit is 6 or greater.

  11. The probability that BOTH events A and B occur • A and B are the overlapping area common to both A and B • Only for INDEPENDENT events

  12. Example If the chances of success for surgery A are 85% and the chances of success for surgery B are 90%, what are the chances that both will fail?

  13. Find: P(A and B) =getting an even number that is at least 5 P(A and C) =getting an even number that is at most 3 P(B and C)=getting a number that is at most 3 and at least 5. Venn Diagram: Intersection (“And”/* Rule)

  14. Many people who come to clinics to be tested for HIV don’t come back to learn the test results. Clinics now use “rapid HIV tests” that give a result in a few minutes. Applied to people who don’t have HIV, one rapid test has probability about .004 of producing a false-positive. If a clinic tests 200 people who are free of HIV antibodies, what is the probability that at least one false positive will occur? N = 200 P(positive result) =.004, so P(negative result)=1-.004=.996 Finding the probability of “at least one”P(at least one) = 1-P(none)

  15. Big Picture • + Rule holds if A and B are disjoint/mutually exclusive • * Rule holds if A and B are independent • * Disjoint events cannot be independent! Mutual exclusivity implies that if event A happens, event B CANNOT happen.

  16. Find: P(A given C) =getting an even number GIVEN that the number is at most 3. P(A given B) =getting an even number GIVEN that the number is at least 5. Conditional probability: Pre-set condition (“given”)

  17. In building new homes, a contractor finds that the probability of a home-buyer selecting a two-car garage is 0.70 and selecting a one-car garage is 0.20. (Note that the builder will not build a three-car or a larger garage). • What is the probability that the buyer will select either a one-car or a two-car garage? • Find the probability that the buyer will select no garage. • Find the probability that the buyer will not want a two-car garage.

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