Came across an interesting problem today. You are given a coin and x money, you double money if you get heads and lose half if tails on any toss.
- What is the expected value of your money in n tries
- What is the probability of getting more than expected value in (1)
This is how I approached it. The probability of heads and tails is same (1/2). Expected value after first toss = 1/2(2*x) + 1/2(1/2*x) = 5x/4 So expected value is 5x/4 after first toss. Similarly repeating second toss expectation on 5x/4, Expected value after second toss = 1/2(2*5x/4) + 1/2(1/2*5x/4) = 25x/16
So you get a sequence of expected values: 5x/4, 25x/16, 125x/64, ...
After n tries, your expected value should be 5^n/4^n * x.
If n is large enough, your expected value should approach the mean of the distribution. So probability that value is greater than expected value should be 0.5. I am not sure about this one.