Warm-Up:1) Define sample space.2) Give the sample space for the sum of the numbers for a pair of

dice.

3) You flip four coins. What’s the probability of getting exactly two heads? (Hint: List the outcomes first).

4) 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

1) Find the probability that the student is not in the traditional undergraduate age group of 18-23

2) Find P(30+ years)

Age group (yr)

18-23 24-29 30-39 40+

Probability .57 .17 .14 .12

Venn Diagram

Find P(A), P(B), P(C)

Find P(A’), P(B’), P(C’)

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

Find:1) P(AUB) =getting an even

number or a number greater than or equal to 5 or both

2) P(AUC) =getting an even number or a number less than or equal to 3 or both

3) P(BUC)=getting a number that is at most 3 or at least 5 or both.

Remember this?

Ex. 6.14, p. 420

• Now find P(rolling a 7) =

• Because all 36 outcomes together must have probability 1 (Rule 2), each outcome must have probability 1/36.

Consider the events A = {first digit is 1}, B = {first digit is 6 or greater}, and C = {a first digit is odd}

a) Find P(A) and P(B)b) Find P(complement of A)c) Find P(A or B)d) Find P(C)e) 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

Ex. 6.16, p. 422

Find the probability of the event B that a randomly chosen first digit is 6 or greater.

1st digit

1 2 3 4 5 6 7 8 9

Prob. 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9

• 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

ExampleIf 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:

1) P(A and B) =getting an even number that is at least 5

2) P(A and C) =getting an even number that is at most 3

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:

1) P(A given C) =getting an even number GIVEN that the number is at most 3.

2) 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).

1) What is the probability that the buyer will select either a one-car or a two-car garage?

2) Find the probability that the buyer will select no garage.

3) Find the probability that the buyer will not want a two-car garage.