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Stat 400, chapter 2, Probability, Conditional Probability and Bayes’ Theorem
supplemental handout prepared by Tim Pilachowski
Example 1. Silver Springs, Florida, has a snack bar and a gift shop. The management observes 100 visitors, and counts 65 who eat in the snack bar. Among those who ate in the snack bar, 40 also make a purchase in the gift shop. Of the patrons who did not eat in the snack bar, 15 bought something in the gift shop.
= = = P SB ′ = − =
N
N SB
P SB
= PG SB
NSB
NG SB
P G SB
′ = P G SB
NSB
NG SB
P G SB
The tree diagram would look like this.
G
SB
G ′
G
20
13
20
7
13
8
13
5
7
3
7
4
G ′
S B ′
What is the probability that a person who made a purchase in the gift shop ate in the snack bar? translation: P (ate in the snack bar | made a purchase) = P ( SB | G ). We need Bayes’ Theorem.
Which branches above make up “event G happens”? the first and third ones down
20
3 20
8
5
2
7
3 20
7 13
8 20
13
13
8 20
13
PSB PG SB P SB PG S B
PSB PG SB
PSB G P SB G
P SB G
P G
PSB G
P SB G
interpretations:
From the given information, P ( G | SB ) = 138 means that 8 out of every 13 people who eat in the snack bar also
made a purchase in the gift shop. From Bayes Theorem P ( SB | G ) = 118 means that 8 out of every 11 people
who make a purchase in the gift shop also eat in the snack bar.
Example 2: The Gallup organization conducted 10 separate surveys conducted from January through May 2009. At the time of the report, Gallup had found an average of 35% of Americans considering themselves Democratic, 37% independent and 28% Republican. Within those affiliations, the following percentages identified themselves as Conservative, Moderate or Liberal.
Democrat (event D ) Independent (event I ) Republican (event R ) Conservative (event C ) 22% 35% 73% Moderate (event M ) 40% 45% 24% Liberal (event L ) 38% 20% 3% http://www.gallup.com/poll/120857/conservatives-single-largest-ideological-group.aspx Results are based on aggregated Gallup Poll surveys of approximately 1,000 national adults, aged 18 and older, interviewed by telephone. Sample sizes for the annual compilations range from approximately 10,000 to approximately 40,000. For these results, one can say with 95% confidence that the maximum margin of sampling error is ±1 percentage point.
This table gives conditional probabilities. P ( Conservative |Independent) = 35 %. P ( Liberal |Republican) = 3 %.
2-01_._ Determine P ( D ), P ( I ) and P ( R ).
2-02. Determine P ( D ′)^ , P ( I ′^ )and P ( R ′)^.
2-03a. Determine P ( M | D ). b. Write a verbal description of what P ( M | D ) means.
2-04a.Write a verbal description of D ∩ M. b. Calculate P ( D ∩ M ).
2-05a. Determine P ( M ′^ | R ). b. Write a verbal description of what P ( M ′^ | R )means.
2-06a. Write a verbal description of R ∩ M ′. b. Calculate P ( R ∩ M ′).
2-07. Draw a tree diagram to illustrate the events and probabilities for this two-stage experiment.
2-08. Calculate P ( M ).
2-09a. Write a verbal description of D ∪ M. b. Calculate P ( D ∪ M ).
2-10a. Use Bayes’ Theorem to calculate P ( D | M ). b. Write a verbal description of what P ( D | M ) means.
2-11. Are D and M independent events? How do you know?
2-12. Calculate P ( M ′).
2-13a. Write a verbal description of R ∪ M ′. b. Calculate P ( R ∪ M ′).
2-14a. Use Bayes’ Theorem to calculate P ( R | M ′). b. Write a verbal description of what P ( R | M ′)means.
2-15. Are R and M ′^ independent events? How do you know?
