A random tangent I have been thinking about, please be gentle I am not a mathematician:
Imagine a coin toss where Heads means we win a coin and Tails means we lose a coin. We start with 1 coin and cannot go negative (i.e. if we flip a Tails on the first try, we lose and it's game over).
The coin is weighted so that it flips Tails with a probability of a. If a = 0, the probability of bankruptcy is 0%. At some value between 0 and 1, the probability of bankruptcy approaches 100%. For what value of a does this probability cross 50%?
So I drew a graph of the outcomes for three flips of a coin with
a = 0.4:
1 (0.4) / \ (0.6) 0 2 (0.4) / \ (0.6) 1 3 / \ / \ 0 2 4 First Flip: P(0) = 40%, P(2) = 60% Second Flip: P(0) = 40%, P(1) = 24%, P(3) = 36% Third Flip: P(0) = 49.6%, P(2) = 28.8%, P(4) = 21.6%
The first thing I noticed is the probability of bankruptcy on evened-numbered flips (e.g. flip #2, flip #4, ...) does not increase. So you have a 40% of busting on the first hand, the probability remains unchanged on the second flip.
I attempted to try and solve this problem in a closed form solution shown below (sorry it's a picture, I don't know LaTeX...).
For a = 1/2, I got a P(0) = 2/3. Obviously this is wrong because for a equally weighted coin, we would expect that playing infinitely long, you would succumb to gambler's ruin pretty quick.
So again restating the original question: How weighted does the coin have to be in our favor that after playing for an arbitrarily long amount of time, our odds of bankruptcy approaches 50%? Is this a problem that can be solved in closed form? Also if you can point out the fallacies in my attempt, I greatly appreciate it.
Thanks in advance!