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## PowerPoint Slideshow about ' Chapters 14, 15 (part 2) Probability Trees, Odds' - penelope-pickett

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### Chapters 14, 15 (part 2) Probability Trees, Odds

Probability Trees: A Graphical Method for Complicated Probability Problems.

Odds and Probabilities

Probability Tree Example: probability of playing professional baseball

- 6.1% of high school baseball players play college baseball. Of these, 9.4% will play professionally.
- Unlike football and basketball, high school players can also go directly to professional baseball without playing in college…
- studies have shown that given that a high school player does not compete in college, the probability he plays professionally is .002.

Question 1: What is the probability that a high school baseball player ultimately plays professional baseball?

Question 2: Given that a high school baseball player played professionally, what is the probability he played in college?

Question 1: What is the probability that a high school baseball player ultimately plays professional baseball

.061*.094=.005734

.939*.002=.001878

P(hs bb player plays professionally)

= .061*.094 + .939*.002

= .005734 + .001878

= .007612

Question 2: Given that a high school baseball player played professionally, what is the probability he played in college?

.061*.094=.005734

.061*.094=.005734

P(hs bb player plays professionally)

= .005734 + .001878

= .007612

.939*.002=.001878

Example: AIDS Testing

- V={person has HIV}; CDC: Pr(V)=.006
- P : test outcome is positive (test indicates HIV present)
- N : test outcome is negative
- clinical reliabilities for a new HIV test:
- If a person has the virus, the test result will be positive with probability .999
- If a person does not have the virus, the test result will be negative with probability .990

Question 1

- What is the probability that a randomly selected person will test positive?

Probability Tree Approach

- A probability tree is a useful way to visualize this problem and to find the desired probability.

Question 1: What is the probability that a randomly selected person will test positive?

Question 2

- If your test comes back positive, what is the probability that you have HIV?

(Remember: we know that if a person has the virus, the test result will be positive with probability .999; if a person does not have the virus, the test result will be negative with probability .990).

- Looks very reliable

Question 2: If your test comes back positive, what is the probability that you have HIV?

Summary

- Question 1:
- Pr(P ) = .00599 + .00994 = .01593
- Question 2: two sequences of branches lead to positive test; only 1 sequence represented people who have HIV.

Pr(person has HIV given that test is positive) =.00599/(.00599+.00994) = .376

Recap

- We have a test with very high clinical reliabilities:
- If a person has the virus, the test result will be positive with probability .999
- If a person does not have the virus, the test result will be negative with probability .990
- But we have extremely poor performance when the test is positive:

Pr(person has HIV given that test is positive) =.376

- In other words, 62.4% of the positives are false positives! Why?
- When the characteristic the test is looking for is rare, most positives will be false.

If event A has probability P(A), then the odds in favor of A are P(A) to 1-P(A). It follows that the odds against A are 1-P(A) to P(A)

If the probability the Boston Red Sox win the World Series is .20, then the odds in favor of Boston winning the World Series are .20 to .80 or 1 to 4. The odds against Boston winning are .80 to .20 or 4 to 1

From Probability to OddsIf the odds in favor of an event E are a to b, then

P(E)=a/(a+b)

If the odds against an event E are c to d, then

P(E’)=c/(c+d)

(E’ denotes the complement of E)

From Odds to ProbabilityE = win World Series

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