Probability 4/18/12

Probability4/18/12

• Probability• Conditional probability• Disjoint events• Independent events

Section 11.1 (pdf) Professor Kari Lock MorganDuke University

• Project 2 Presentation (TOMORROW)

• Read The Bayesian Heresy, from The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century

• Homework 10 (due Monday, 4/23)

• Project 2 Paper (Wednesday, 4/25)

To Do

Presentations• Each group will get 10 minutes to present, and this will be strictly enforced. Plan on 8 minutes of presentation, 2 minutes for questions.

• No need to present your whole project – choose only whatever is most interesting

• Make sure to upload your presentation slides to Sakai by 8:30 am tomorrow (name: team name)

• Presentation order will be randomly determined and we will start on time, so make sure to be there on time

• Reminder: all group members must participate! Not participating will result in an individual 0 for the presentation component.

Office Hours Today• 3-5 pm (Professor Morgan - Old Chem 216)

• 7-9 pm (Michael - Old Chem 211A)

PROBABILITY

Event• An event is something that either happens or doesn’t happen, or something that either is true or is not true

• Examples:• Y = 1• Y > 90• Gender = male• It rains today• The Bruins win the Stanley Cup again

Probability• The probability of event A, P(A), is the probability that A will happen

• Probability is always between 0 and 1

• Probability always refers to an event

• P(A) = 1 means A will definitely happen• P(A) = 0 means A will definitely not happen

Probability Examples• Y = number of siblings. P(Y = 1) = 0.481(based on your survey data)

• Y: final grade in STAT 101. P(Y > 90) = 0.338(based on last semester’s class)

• P(Gender = male) = 0.506 (for Duke students, www.usnews.com)

• P(it rains today) = 0.7 (www.weather.com)

• P(The Bruins win the Stanley Cup again) = 1/6 (www.bleacherreport.com)

Sexual Orientation

• What are the sexual orientation demographics of American adults?

• We need data!

• Data collected in 2009 on a random sample of American adults (National Survey of Sexual Health and Behavior)

Male Female TotalHeterosexual 2325 2348 4673Homosexual 105 23 128Bisexual 66 92 158Other 25 58 83Total 2521 2521 5042

Sexual Orientation

Herbenick D, Reece M, Schick V, Sanders SA, Dodge B, and Fortenberry JD (2010). Sexual behavior in the United States: Results from a national probability sample of men and women ages 14–94. Journal of Sexual Medicine;7(suppl 5):255–265.

Male Female TotalHeterosexual 2325 2348 4673Homosexual 105 23 128Bisexual 66 92 158Other 25 58 83Total 2521 2521 5042

Sexual Orientation

What is the probability that an American adult is homosexual?

a) 128/5042 = 0.025b) 128/4673 = 0.027c) 105/2521 = 0.04d) I got a different answer

Two Events

• P(A and B) is the probability that both events A and B will happen

• P(A or B) is the probability that either event A or event B will happen

B

A and B

A

A or B: all color

Two Events

Male Female TotalHeterosexual 2325 2348 4673Homosexual 105 23 128Bisexual 66 92 158Other 25 58 83Total 2521 2521 5042

Sexual Orientation

What is the probability that an American adult is male and homosexual?

a) 105/128 = 0.82b) 105/2521 = 0.04 c) 105/5042 = 0.021d) I got a different answer

Male Female TotalHeterosexual 2325 2348 4673Homosexual 105 23 128Bisexual 66 92 158Other 25 58 83Total 2521 2521 5042

Sexual Orientation

What is the probability that an American adult is female or bisexual?

a) 2679/5042 = 0.531b) 2587/5042 = 0.513 c) 92/2521 = 0.036d) I got a different answer

P(A or B)

( or ) ( ) ( ) ( and )P A B P A P B P A B

B

A

Male Female TotalHeterosexual 2325 2348 4673Homosexual 105 23 128Bisexual 66 92 158Other 25 58 83Total 2521 2521 5042

Sexual Orientation

What is the probability that an American adult is not heterosexual?

a) 369/5042 = 0.073b) 2587/5042 = 0.513 c) 92/2521 = 0.036d) I got a different answer

P(not A)

(not ) 1 ( )P A P A

CaffeineBased on last year’s survey data, 52% of students drink caffeine in the morning, 48% of students drink caffeine in the afternoon, and 37% drink caffeine in the morning and the afternoon. What percent of students do not drink caffeine in the morning or the afternoon?

a) 63%b) 37%c) 100%d) 50%

Conditional Probability• P(A if B) is the probability of A, if we know B has happened

• This is read in multiple ways:

• “probability of A if B”• “probability of A given B” • “probability of A conditional on B”

• You may also see this written as P(A | B)

Male Female TotalHeterosexual 2325 2348 4673Homosexual 105 23 128Bisexual 66 92 158Other 25 58 83Total 2521 2521 5042

Sexual Orientation

What is the probability that an American adult male is homosexual?

a) 105/128 = 0.82b) 105/2521 = 0.04 c) 105/5042 = 0.021d) I got a different answer

Male Female TotalHeterosexual 2325 2348 4673Homosexual 105 23 128Bisexual 66 92 158Other 25 58 83Total 2521 2521 5042

Sexual Orientation

What is the probability that an American adult homosexual is male?

a) 105/128 = 0.82b) 105/2521 = 0.04 c) 105/5042 = 0.021d) I got a different answer

Conditional Probability

P(homosexual if male) = 0.04

P(male if homosexual) = 0.82

( if f ) ( i )P P BA B A

Conditional Probability

( if ) ( and )( )

P AP BAP

BB

AB

CaffeineBased on last year’s survey data, 52% of students drink caffeine in the morning, 48% of students drink caffeine in the afternoon, and 37% drink caffeine in the morning and the afternoon. What percent of students who drink caffeine in the morning also drink caffeine in the afternoon?

a) 77%b) 37%c) 71%

Helpful TipIf the table problems are easier for your than the sentence problems, try to first convert what you know into a table.

52% of students drink caffeine in the morning, 48% of students drink caffeine in the afternoon, and 37% drink caffeine in the morning and the afternoon Caffeine

AfternoonNo Caffeine Afternoon

Total

Caffeine MorningNo Caffeine MorningTotal 100

52

48

3748

52

153711

P( afternoon if morning) = 37/52 = 0.71

P(A and B)

( if ) (( and ) )P P A BA PB B

( if ) ( and )( )

P AP BAP

BB

Duke Rank and Experience60% of STAT 101 students rank their Duke experience as “Excellent,” and Duke was the first choice school for 59% of those who ranked their Duke experience as excellent. What percentage of STAT 101 students had Duke as a first choice and rank their experience here as excellent?

a) 60%b) 59%c) 35%d) 41%

( or ) ( ) ( ) ( and )P A B P A P B P A B

( if f ) ( i )P P BA B A

Summary

( if ) (( and ) )P P A BA PB B

(not ) 1 ( )P A P A

( if ) ( and )( )

P AP BAP

BB

Disjoint Events• Events A and B are disjoint or mutually exclusive if only one of the two events can happen

• Think of two events that are disjoint, and two events that are not disjoint.

Disjoint Events

If A and B are disjoint, then

a) P(A or B) = P(A) + P(B)

b) P(A and B) = P(A)P(B)

P(A or B)

( or )( ) ( ) ( and )P A BP A P B P A B

B

A

SPECIAL CASE:If and are disjoint:

( o ) ) ( )r (P BP B AB

A PA

B

A

Independence• Events A and B are independent if P(A if B) = P(A).

• Intuitively, knowing that event B happened does not change the probability that event A happened.

• Think of two events that are independent, and two events that are not independent.

Independent Events

If A and B are independent, then

a) P(A or B) = P(A) + P(B)

b) P(A and B) = P(A)P(B)

P(A and B)

( if ) (( and ) )P P A BA PB B

If and are independent,then ( if ) ( ), soA BP A B P A

SPECIAL CASE: If and are independent,

( and ) ( ) ( )A B

PA AB PP B