# 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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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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