perplexus.info :: Probability : 2 coins

2 coins (Posted on 2016-03-18)

Suppose you have 2 identical-looking coins: one that is fair and one that comes up heads 75% of the time.
You randomly chose one of them, flip it 3 times and it comes up as H-H-T.

What is the chance that the coin you picked is the fair one?

Source: Project Euler

No Solution Yet Submitted by Ady TZIDON

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Bayes

| Comment 2 of 8 |

The probability that you would pick the fair coin and it would come up HHT is 1/2 * 1/8 = 1/16.

The probability that you would pick the biased coin and it would come up HHT is 1/2 * (3/4)^2 * 1/4 = 9/128.

By Bayes' rule/theorem, the probability that you did pick the fair coin given this outcome, is (1/16) / (1/16 + 9/128) = 8/17.

or, in 128 trials with exact regard to probabilities, 8 out of 17 cases of HHT were from the fair coin:

fair HHH

fair THH

fair HHH

fair TTH

fair HTH

fair TTH

fair HTH

fair THH

fair HHH

fair THH

fair HHH

fair TTH

fair HTT

fair TTT

fair HTT

fair THT

fair HHT *

fair THT

fair HHT *

fair TTT

fair HTT

fair TTT

fair HTT

fair THT

fair HHT *

fair THT

fair HHT *

fair TTT

fair HTH

fair TTH

fair HTH

fair THH

unfair HHH

unfair THH

unfair HHH

unfair THH

unfair HHH

unfair HTH

unfair TTH

unfair HTH

unfair HHH

unfair THH

unfair HHH

unfair THH

unfair HHH

unfair THH

unfair HHH

unfair HTH

unfair TTH

unfair HTH

unfair HHH

unfair THH

unfair HHH

unfair THH

unfair HHH

unfair THH

unfair HHT *

unfair HTT

unfair TTT

unfair HTT

unfair HHT *

unfair THT

unfair HHT *

unfair THT

unfair HHT *

unfair THT

unfair HHH

unfair HTH

unfair TTH

unfair HTH

unfair HHH

unfair THH

Posted by Charlie on 2016-03-18 11:10:23

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