I randomly chose between 2 coins. One of the coins has a 0.8 chance of heads and a 0.2 chance of tails. The other is a fair coin that has a 0.5 chance of either heads or tails.
I flip this coin twice and get 2 heads. What's the probability that my next flip is heads?
I tried using Bayes' Theorem:
$$
P(H|HH) = \frac{P(HH|H)P(H)}{P(HH)}
$$
But then $P(HH|H)$ is not easy to solve…
Best Answer
Let $HH$ be the event that we get $2$ heads in a row. Let $H_3$ be the event the third toss is a head. We want $\Pr(H_3|HH)$. Maybe we start from something a little simpler than the Bayes' Theorem that you used, essentially the definition of conditional probability: $$\Pr(H_3|HH)=\frac{\Pr(H_3 \cap HH)}{\Pr(HH)}.$$
We calculate the two probabilities on the right. For $\Pr(HH)$, note that two heads in a row happens with probability $(4/5)^2$ if we use the funny coin, and with probability $(1/2)^2$ if the coin is the ordinary coin. It follows that $$\Pr(HH)=\frac{1}{2}\left(\frac{4}{5}\right)^2+\frac{1}{2}\left(\frac{1}{2}\right)^2.$$
A similar calculation gives us the probability of $HH$, followed by $H_3$: $$\Pr(HH\cap H_3)=\frac{1}{2}\left(\frac{4}{5}\right)^3+\frac{1}{2}\left(\frac{1}{2}\right)^3.$$ Divide.