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?

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.

Disjoint/Complement

Probability Distribution Age group (yr) 18-23 24-29 30-39 40+ Probability .57 .17 .14 .12 Find the probability that the student is not in the traditional undergraduate age group of 18-23 Find P(30+ years)

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

Venn Diagram: Union (“Or”/Addition Rule) 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.

Remember this?

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) =

Find P(complement of A) Find P(A or B) Find P(C) Find P(B or C) 1st digit 1 2 3 4 5 6 7 8 9 Prob. .301 .176 .125 .097 .079 .067 .058 .051 .046 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)

Ex. 6.16, p. 422 1st digit 1 2 3 4 5 6 7 8 9 Prob. 1/9 Find the probability of the event B that a randomly chosen first digit is 6 or greater.

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

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?

Venn Diagram: Intersection (“And”/* Rule) 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.

Finding the probability of “at least one” P(at least one) = 1-P(none) 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

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.

Conditional probability: Pre-set condition (“given”) 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.

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.