I have the calculations
$$P(S_3=S_2)=\frac{P(S_2=2\mid S_3=3)P(S_3=3)}{\sum^3_{i=0}P(S_2=2\mid S_3=i)P(S_3=i)} = \frac{1\cdot1/8}{1/2\cdot3/8 + 1\cdot1/8}=0.4$$
I know the answer is $1/2$ which is not what I get. My problem is calculating
$$P(S_2=2\mid S_3=2)$$
I use my intuition to get $1/2$ but clearly my intuition is wrong. The above is the classic experiment of tossing a coin hence $S$ is binomial.
Can someone please help explain why my intuition is wrong and how one can calculate this? I can't seem to put it in a Bayesian framework.