C HAPTER 15 Probability Rules! "We figured the odds as best we could, and then we rolled the dice." U.S. President Jimmy Carter, June 10,1976.

CHAPTER 15

Probability Rules!

"We figured the odds as best we could, and then we rolled the dice."U.S. President Jimmy Carter, June 10,1976

Useful tools for Probability1) Venn Diagram: sometimes helpful, especially when given the probability of separate events, e.g., P(A) and P(B) as well as the joint probability P(A and B).2) Two-way (contingency) table: useful for interpreting conditional probabilities and examining independence. Each cell is the joint probability for the row and column events that define the cell. The total probability for all cells must be 1, and the sum for each row (column) gives the “marginal” probabilities for that row (column).3) Tree Diagram: A display of conditional events or probabilities that is helpful in finding the probability that two or more events occur together. Simply multiply along the branches that correspond to the outcomes of interest. It provides easy-to-interpret models for situations in which a scenario may be decomposed into multiple stages, one following another.

Example 2: Police report that 78% of drivers stopped on suspicion of drunk driving are given a breath test, 36% a blood test, and 22% both tests. What is the probability that a randomly selected DWI suspect is givena) a test?b) a blood test or a breath test, but not both tests?c) neither test?d) Are blood test and breath test mutually exclusive?

Show: Venn Diagram P(A) = P(breath test); P(B) = P(blood test)

0.56A 0.14B 

Drawing w/o replacement changes probability of the next card drawn

Example: I draw one card and look at it and tell you it is red. What is the probability it is a heart? Then what is the probability that it is red, given that it is a heart?

Think: The card is selected at random.

Show: P(heart|red) =

P(heart and red)/P(red) = (13/52)/(26/52) = ½

P(red|heart) = P(red and heart)/P(heart) = (13/52)/(13/52) = 1

Tell: If a randomly chosen card is a red card, the probability that it is a heart is 0.5. If a randomly chosen card is a heart, it is certain to be red.

Disjoint (mutually exclusive events) can’t be independent events.

Disjoint events have ______________ in common, so knowing that one occurred means the other did not occur. Don’t make the mistake of saying that they are the same!

Example: Are “red card” and “spade” independent or mutually exclusive?

Example: Are “red card” and “ace” independent or mutually exclusive?

Example: Are “face card” and “king” independent or mutually exclusive?

Mutually exclusive; spade can’t be red

  Jeans Other TotalMale 12 5 17Female 8 11 19Total  20 16 36

Two-Way (Contingency) Table: Given the table of people who wear jeans

What is the probability that a male wears jeans? P(jeans|male) = 12/17

What is the probability that someone wearing jeans is a male?

P(male|jeans) = 12/20 Are being male and wearing jeans disjoint?

Are gender and attire independent?

Tree Diagrams: Example: Video-sharing sites, led by YouTube, are popular destinations on the Internet. Looking at users aged 18 and over, 27% are 18 to 29, another 45% are 30-49, and the remaining 28% are 50 and over. The Pew Internet and American Life Project finds that 70% of Internet users aged 18 to 29 have visited a video-sharing site, along with 51% of those aged 30 to 49, and 26% of those 50 or older. Do most Internet users visit YouTube and similar sites?

Think: What percent of all adult Internet users visit video-sharing sites? 

Tree Diagrams: Example (continued): Show:   

18 to 29

30 to 49

50+

Adult internet users

0.7

0.30.51

0.49

0.26

0.74

Video Yes: 0.1890

Video No: 0.0810

Video Yes: 0.2295

Video No: 0.2205

Video No: 0.2072

Video Yes: 0.07280.28

0.27

0.45

P(18 to 29) * P(video yes | 18 to 29)

Tree Diagrams: Example (continued): Show: P(video yes) = 0.1890 + 0.2295 + 0.0728 = 0.4913  Tell: About 49% of all adult Internet users have visited a video-sharing site. Reversing the Condition: What percent of adult Internet users who visit video-sharing sites are aged 18 to 29?Show: P(18 to 29) | video yes) = P(18 to 29 and video yes)

P(video yes)

= 0.1890 = 0.3847

0.4913

About 38% of adults who visit video-sharing sites are between 18 and 29.