You start with 100 dollars. When you flip a coin and receive a head, you gain 100 dollars. If you flip the coin and received a tail, you lose half of your current money. What is the expected amount of money you will have after 4 flips?
I know that if you gained 100 dollars on heads and lost 50 dollars on tails you could say: E(final money) = (100 * 0.5) – (50 * 0.5).
I also know that if you doubled your money on heads and halved your money on tails then E(final money) = 100 * 2^(# of heads) * (1/2)^(#of tails)
However, this problem seems to be mixing both addition and multiplication, so I am unsure of how to combine them into a single equation to receive a solution.
Best Answer
Let $E_n$ be the expected money after n flips. Using recursion,
$$E_{n+1}=\frac12(\frac12E_{n}+(E_n+100))=\frac34E_n+50$$
$E_1=125$
$E_2=\frac{375}4+50=\frac{575}4$
$E_3=\frac{3\cdot575}{16}+50=\frac{2525}{16}$
$E_4=\frac{3\cdot2525}{64}+50=\frac{10775}{64}=168\frac{23}{64}$
Therefore your money increase by $68\%$. $:)$