Lets take two coins, for example. One is biased at 60 percent heads, the other is fair. Lets say we toss them 4 times, and we desire the string HTHT.
I have a homework question to determine the joint probability that we selected the fair coin at random, and then proceeded to flip it 4 times to give the string HTHT.
I jumped to Bayes theorem for this question; The probability of getting the string HTHT given that we had selected the fair coin. I know the following.
The probability of selecting either coin is 0.5.
If we knew we had selected the fair coin, then our probability of
flipping HTHT is 2^(-4)
If we knew we had selected the biased coin, then our probability of
flipping HTHT is 0.6^2 * 0.4^2 == 0.0576
This means the total probability of getting the string HTHT from the
moment we begin selecting our coin to when we're done flipping is
2^(-4) + 0.0576 == .1201
Plugging this into Bayes gives
P(HTHT | fair coin) == (.1201 * P(fair coin | HTHT) ) / 0.5
The problem I'm having with Bayes theorem in this approach is that I need to know the probability of having selected the fair coin given that my string was HTHT.
How would I determine that?
I'm stupid.
I would take the probability of flipping HTHT if I knew for certain I had the fair coin, and then divide it by the total probability of flipping HTHT given either coin.
This comes out to
2^(-4) / 0.1201 ~= 0.52