Which coin was it?
A box holds two coins that look the same: one is fair, the other has heads on both sides. You take one at random and toss it three times. It shows heads all three times.
What is the probability that you hold the two-headed coin?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Bayes' rule: compare the probability of three heads under each coin, weighted by the prior .
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Both coins were equally likely before the tosses, so the posterior odds are just the ratio of the likelihoods of the observed data.
Step 1: likelihoods. The two-headed coin gives three heads with probability . The fair coin gives three heads with probability .
Step 2: Bayes.
In odds form. Prior odds , likelihood ratio , so posterior odds , which is a probability of . Each additional head doubles the odds.