Conditional expectation given two outcomes

Question

The solution for the question$^1$ is $E(S_3|S_1(H)) = p^2(HHH) + pq(HHT) + qp(HTH) + q^2(HTT)$; but what is it to $E(S_3|S_1)$?

Probably shift one node up and then we have p^3 and q^3 but is then pq^2, qp^2?

1: Stochastic Calculus For Finance: Conditional Expectation of Binomial Tree Model

Answer 1

Using the notation of the linked question:$$\mathbb E(S_3\mid S_1)=\mathbb E(S_3\mid S_1(\mathsf H))\,\mathbf 1_{S_1(\mathsf H)}+\mathbb E(S_3\mid S_1(\mathsf T))\,\mathbf 1_{S_1(\mathsf T)}$$

The solution for the question$^1$ is $${\mathbb E(S_3\mid S_1(\mathsf H)) ~{= p^2S_3(\mathsf {HHH}) + pqS_3(\mathsf {HHT}) + qpS_3(\mathsf {HTH}) + q^2S_3(\mathsf {HTT})\\=32p^2+16pq+2q^2}}$$

Indeed. The coefficients are the conditional probability for the string of three coin flips when given that it starts with one head showing.

Similarly the conditional expectation for the value of three coin flips when given the value of the first flip indicates that it is a tail, is going to have the same coefficients (since each subsequent coin flip is independent of the previous) and differ only by relevant the valuations:-

$${\mathbb E(S_3\mid S_1(\mathsf T)) ~{= p^2S_3(\mathsf {THH}) + pqS_3(\mathsf {THT}) + qpS_3(\mathsf {TTH}) + q^2S_3(\mathsf {TTT})\\=8p^2+4pq+0.50q^2}}$$

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With the valuations taken from the diagram below, which indicates you start with $4$, double for each head showing, and halve for each tail. Ie: let $h$ be the count for heads in a string of length $h+t$, then the string has the value $2^{2+h-t}$.