True and False Positives

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Problem 1:

Expensive Guitar Ownership as

an Indication of Guitar Virtuosity

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Problem 1

  • 1 out of every 5000 guitar owners is a guitar virtuoso.
  • 98 percent of guitar virtuosos own a guitar worth over $10,000.
  • 20 percent of non-virtuoso guitar owners own a guitar worth over $10,000.

QUESTION: If Bob owns a guitar worth over $10,000, what is the probability that Bob is a guitar virtuoso?

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Solution

  • "1/5000 guitar owners is a virtuoso" gives us a ratio of 1:4999. That's 1 virtuoso to 4999 non-virtuosos.
  • Calculate the number of true positives (owners of guitars worth over $10,000 who are virtuosos):
    • 1 x 0.98 = 0.98 true positives
  • Calculate the number of false positives (owners of guitars worth over $10,000 who are not virtuosos)
    • 4999 x 0.20 = 9998 false positives

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Solution

So there are 0.98 true positives out of a total of

0.98 + 999.8 = 1000.78 positives.

(positives = owners of guitars worth over $10,000)

 

prob(Bob is a virtuoso) = 0.98/1000.78 = 0.00098

= not very likely

 

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Problem 2

Sports Car Ownership as

an Indication of Middle-Aged Insecurity

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Problem 2

  • 1 out of every 20 middle-aged men is emotionally insecure.
  • 80 percent of emotionally insecure middle-aged men drive sports cars.
  • 2 percent of emotionally secure middle-aged men drive sports cars.

QUESTION: If Bob is a middle-aged man who drives a sports car, what is the probability he's emotionally insecure?

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Solution

  • The ratio is 1:19. That's 1 insecure middle-aged man to 19 secure middle-aged men.
  • Calculate true positives:
    • 1 x 0.8 = 0.80 (insecure middle-aged men with sports cars)
  • Calculate false positives:
    • 19 x 0.02 =    0.38 (secure middle-aged men with sports cars)

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Solution

So there are 0.80 true positives and 0.38 false positives.

 

prob(Bob is emotionally insecure) =

0.80/(0.80 + 0.38) = 0.67797

= about a 2 out of 3 chance

 

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Problem 3

Coolness as 

an Indication of Preference for Coke

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Problem 3

  • 5 out of 12 Texans prefer Coke to Dr. Pepper.
  • 95 percent of Coke drinkers are cool.
  • 5 percent of Dr. Pepper drinkers are cool.

QUESTION: If Bob is cool, what is the probability that he drinks Coke?

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Solution

  • The ratio is 5 to 7. That's 5 Coke drinkers to 7 Dr. Pepper drinkers.
  • Calculate the true positives (cool people who drink Coke):
    • 5 x 0.95 = 4.75
  • Calculate the false positives (cool people who drink Dr. Pepper):
    • 7 x 0.05 = .35

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Solution

So there are 4.75 + 0.35 = 5.10 positives (cool people).

 

prob(Bob drinks Coke) = 4.75/5.10 = 0.93137

= Cool Texans drink Coke

 

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True and False Positives Problems

By smilinjoe

True and False Positives Problems

a.k.a., Conditional probability problems: If A is true, what's the probability of B?

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