Suppose I have either a fair coin or a bent coin, and I don’t know which. The bent coin has a 60%60\%
60% probability of coming up heads.
I throw the coin ten times and it comes up heads 8 times. What is the probability I have the fair coin vs. the probability I have the bent coin? Assume at the outset there is an equal (.5,.5)(.5,.5)
(.5,.5) prior probability of either coin.
*Please note that in order to fit the entire formula in the feedback, probability has been abbreviated to "prob."
Answers
Given : either a fair coin or a bent coin bent coin has a 60% probability of coming up heads. throw the coin ten times and it comes up heads 8 times.
To find : probability fair coin vs. probability bent coin
Solution:
if its Fair coin
then probability of Head Coin = 1/2
Probability of tail Coin = 1/2
Probability of head 8 times out of 10
= ¹⁰C₈(1/2)⁸(1/2)²
= 45/1024
= 0.0439
if its bent coin
then probability of Head Coin = 0.6 = 6/10 = 3/5
Probability of tail Coin = 1 - 3/5 = 2/5
Probability of head 8 times out of 10
= ¹⁰C₈(3/5)⁸(2/5)²
= 45 * 6561 * 4 / 5¹⁰
= 0.1209
Probability of Fair coin = 0.0439 /(0.0439 + 0.1209) = 0.2664
Probability of bent coin = 0.1209 /(0.0439 + 0.1209) = 0.7336
Probability of Fair coin = 26.64 %
Probability of Bent coin = 73.36 %
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