Which coin did I grab? A Bayesian posterior

Question

A bag holds two coins: a fair coin with $P(\text{heads}) = 0.5$ and a trick coin with $P(\text{heads}) = 0.8$ . You grab one at random (equal chance of either) and flip it 3 times, getting heads all three times.

Using Bayes' rule, what is the probability that you grabbed the trick coin?

Accepted Answer

Intuition first. Both coins can produce three heads, but the trick coin does it far more easily, so a run of heads is evidence that you are holding the trick coin. Bayes' rule turns that gut feeling into a number by weighing "how likely was this data under each coin," starting from the even 50/50 prior. The recipe. For each hypothesis, multiply its prior probability by the chance it would produce the data. Then divide each product by the total to get the updated (posterior) probabilities. Chance of three heads under each coin: P(	ext{HHH}\mid	ext{fair}) = 0.5^3 = 0.125, \qquad P(	ext{HHH}\…

Which coin did I grab? A Bayesian posterior

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