# Chapters 14, 15 (part 2) Probability Trees, Odds - PowerPoint PPT Presentation

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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.

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

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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.

### Probability Tree

Multiply

branch probs

clinical reliability

clinical reliability

### 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

### 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.

## ODDS AND PROBABILITIES

World Series Odds

From probability to odds

From odds to probability

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 Odds

If 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 Probability

E = win World Series