I'm gambling with a $\$100$ entry fee. Heads gives me $+30\%$ gain, and tails gives me $-25\%$. I decide to pull out after $2$ flips, so the function should be written as something like:
$$X \sim B\left(2, \frac{1}{2} \right)$$
Except that the successes/fails don't quite give the same opposite results. How do I write this binomial function with $X$ as the value of my wallet after $2$ flips? Is this even a binomial distribution?
It’s the number of heads that’s binomially distributed according to $N\sim B\left(2,\frac12\right)$. The number of your dollars at the end of the experiment is $X=100\cdot 1.3^N0.75^{2-N}$.