Fun math problem...Bayesian
August 1st, 2017 at 9:27:16 PM
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A New York City cab was involved in a hit-and-run accident last night. Five witnesses reported the incident, four of whom said that the cab was green and one of whom said that the cab was yellow. Assume each witness correctly identifies the color of a cab with probability 2/3. It is known that 85% of registered cabs in New York City are yellow and 15% are green. Based on this information, what is the probability that the cab was green?
August 1st, 2017 at 10:58:23 PM
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Quote: Ace...Based on this information, what is the probability that the cab was green?
only 58.54%.
Solve this equation for the first probability:
P(it was green if 4 say green, 1 says yellow) * P( 4 say green, 1 says yellow) = P(a taxi is green) * P ( 4 say green, 1 says yellow if the taxi is green)
Solve this equation for the first probability:
P(it was green if 4 say green, 1 says yellow) * P( 4 say green, 1 says yellow) = P(a taxi is green) * P ( 4 say green, 1 says yellow if the taxi is green)
August 2nd, 2017 at 6:07:17 AM
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There are two possibilities: either the cab was green, and 4 out of the 5 witnesses were correct, or the cab was yellow, and only 1 of the 5 was correct.
P(green) = 3 / 20 x 5 x (2 / 3)4 x 1 / 3 = (15 x 16) / (20x 243) = 240 / (20 x 243)
P(yellow) = 17 / 20 x 5 x (1 / 3)4 x 2 / 3 = (85 x 2) / (20 x 243) = 170 / (20 x 243)
P(green) / (P(green) + P(yellow)) = 240 / (240 + 170) = 24 / 41
In other words, what ChesterDog said.
August 2nd, 2017 at 6:21:23 AM
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Classic Bayesian problem. I see a couple right answer above me so I'll be brief. Here is my answer:
0.15*5*(2/3)^4*(1/3)/(0.15*5*(2/3)^4*(1/3)+0.85*5*(2/3)^1*(1/3)^4) = 0.585365854
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
August 2nd, 2017 at 8:02:22 AM
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Right, right and right.
August 2nd, 2017 at 8:48:28 AM
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Since this is done and solved (multiple times over)... 4/5 witnesses said the cab was green. Thus, the cab was green or the people of NY desperately need a revamped educational system to work on their colors.
Playing it correctly means you've already won.
August 2nd, 2017 at 8:51:22 AM
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Quote: Romes...or the people of NY desperately need a revamped educational system to work on their colors.
This....because the cab was clearly white! 😁
DUHHIIIIIIIII HEARD THAT!
