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.

  • 2
    $\begingroup$ There are so few scenarios that you can just do the computation by hand. $\endgroup$
    – lulu
    Sep 14, 2018 at 0:16
  • $\begingroup$ There are 16 possibilities in total so just calculate each case. $\endgroup$
    – abc...
    Sep 14, 2018 at 0:19
  • 1
    $\begingroup$ If $\mu_n$ is the expected money after $n$ flips, then $\mu_{n+1} = \frac{1}{2} (\mu_n + 100) + \frac{1}{2} (\frac{1}{2} \mu_n)$. $\endgroup$ Sep 14, 2018 at 0:22

2 Answers 2


Let $E_n$ be the expected money after n flips. Using recursion,






Therefore your money increase by $68\%$. $:)$


Let's call $X_n$ a random variable that represents the number of money after $n$ coin flips. We have $$ \mathbb{E}[X_{n+1} | X_n] = \frac{X_n+100}{2} + \frac{X_n}{4} = \frac{3X_n}{4} + 50$$ Moreover $$\mathbb{E}[X_{n+1}] = \mathbb{E}[\mathbb{E}[X_{n+1}|X_n]] = \frac{3}{4}\mathbb{E}[X_n] + 50$$ (I used standard properties of Conditional Expectation)


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