<s>[INST] gambling probability problem We are given a fair coin. We start out with 5 dollars. We keep tossing the coin. If the outcome is different than the previous one, we are awarded another 5 dollars. However, we do not get anything if the outcome is the same as the previous one. Let's say we toss the coin X times in the long run. How much do we expect to have in the end?
 [/INST] This problem may be solved using generating functions which is admittedly more powerful than strictly necessary in this case, but it does illustrate the method, which is worth knowing. Let $p_A(n)$ be a polynomial in $u$ such that $[u^k] p_A(n)$ is the probability of having a capital of $k$ dollars and the last coin came up heads and similarly $p_B(n)$ for tails. Then we have the following system of equations
$$ p_A(n) = \frac{1}{2} p_A(n-1) +  \frac{1}{2} u^5 p_B(n-1) \\
 p_B(n) = \frac{1}{2} u^5 p_A(n-1) +  \frac{1}{2} p_B(n-1)$$
with initial conditions
$$ p_A(0) = \frac{1}{2} u^5 \quad \text{and} \quad p_B(0) = \frac{1}{2} u^5.$$ 
Now introduce the generating functions
$$ P_A(z) = \sum_{n\ge 0} p_A(n) z^n  \quad \text{and} \quad 
P_B(z) = \sum_{n\ge 0} p_B(n) z^n.$$
Multiply the recurrences by $z^n:$
$$ p_A(n) z^n = z \frac{1}{2} z^{n-1} p_A(n-1) + z \frac{1}{2} u^5 z^{n-1} p_B(n-1) \\
 p_B(n) z^n = z \frac{1}{2} u^5 z^{n-1} p_A(n-1) + z \frac{1}{2} z^{n-1} p_B(n-1)$$
Sum over $n$ to obtain the following system of equations
$$ P_A(z) -  \frac{1}{2} u^5 = \frac{1}{2}z P_A(z) + \frac{1}{2}u^5 z P_B(z) \\
P_B(z)  -  \frac{1}{2} u^5 = \frac{1}{2}u^5 z P_A(z) + \frac{1}{2}z P_B(z).$$
The solution to this system is
$$ P_A(z)=-{\frac {{u}^{5}}{{u}^{5}z-2+z}}\\
P_B(z) =-{\frac {{u}^{5}}{{u}^{5}z-2+z}}.$$
The generating function for the expected capital after $n$ steps is then given by
$$ f(z) = \left( \frac{d}{du} (P_A(z) + P_B(z)) \right)_{u=1} =
-10\, \left( 2\,z-2 \right) ^{-1}+10\,{\frac {z}{ \left( 2\,z-2 \right) ^{2}}}.$$
Now we have $$[z^n] f(z) = 5+5/2\,n,$$
which is precisely what was obtained above.
We can also compute the variance, which is where the generating functions start to shine. First compute the expected factorial moment $Q(Q-1)$ where $Q$ is the capital. This has generating function
$$ g(z) = \left( \frac{d}{du} \frac{d}{du} (P_A(z) + P_B(z)) \right)_{u=1} =
-40\, \left( 2\,z-2 \right) ^{-1}+140\,{\frac {z}{ \left( 2\,z-2 \right) ^{2}}}-100\,{\frac {{z}^{2}}{\left( 2\,z-2 \right) ^{3}}}
.$$ This gives $$[z^n] g(z) = 20+{\frac {115}{4}}\,n+{\frac {25}{4}}\,{n}^{2}.$$
The variance can now be obtained from 
$$Var[Q] = E[Q^2] - E[Q]^2 = E[Q(Q-1)] + E[Q] - E[Q]^2,$$
which gives
$$Var[Q] = {\frac {25}{4}}\,n.$$
We could in fact compute all factorial moments if desired.
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