Suppose we are given a "radioactive" coin. The coin is such that the probability of obtaining a head drops to half its current value each time a head is obtained. If a tail is obtained, the probability remains the same.
Let $P_n(H)$ denote the probability of obtaining a head after $n$ tosses. Also assume that at the beginning of the experiment, $n = 0$ and $P_0(H) = \dfrac{1}{2}$ .
Its clear that
$P_n(H) = \left[\dfrac{P_{n-1}(H)}{2} \times P_{n-1}(H)\right] + \left[P_{n-1}(H) \times (1 - P_{n-1}(H))\right]$
because
$P(H \text{ on nth toss} ) = P(H \text{ on nth toss} |H \text{ on (n-1) th toss} ) P(H \text{ on (n-1) th toss} ) + P(H \text{ on nth toss} |T \text{ on (n-1) th toss} ) P(T \text{ on (n-1) th toss} ) $
This gives us
$P_n(H) = P_{n-1}(H) - \dfrac{(P_{n-1}(H))^2}{2}$
I was wondering how this pans out, is there a closed form solution etc.