Brainteaser - Flipping Coin to Maximise Ratio

D-Generate

Hi All,

This is a quick brainteaser/interview question of which I can't figure out the answer. Hopefully one of you can shed some light on the below

If you flip a coin until you decide to stop and you want to maximize the ratio of heads to total flips, what is that expected ratio?



I have a hunch that it converges to around the region of 0.61 or so but i feel my logic is flawed.

I drew a lattice and terminated the position if your ratio of heads to tosses was greater than 0.5 and also if your next toss would have an expected loss of ratio.
For example,
At toss 3, if you currently have 2 heads then your ratio is 2/3 so you are greater than 0.5. The benefit of tossing again is 0.5(3/4-2/3)-0.5(2/3-2/4) = -0.04 so no benefit in tossing again and as such this is a termination point.

So then the formula worked out to be approaching
1/2(1)+ (1/2)^3*2/3+(1/2)^5*3/5+(1/2)^7*4/7+(1/2)^9*5/9...

RoundTower

0.61 can't be right, here's a strategy that gets 0.75:

If you get heads first throw, stop. (prob 0.5, ratio 1)
If you get tails then heads, stop. (prob 0.25, ratio 0.5)
If you get tails then tails, go again a million times, your ratio will eventually go to 0.5 (prob 0.25, ratio 0.5)

This strat obviously isn't optimal either.

The benefit of tossing again is 0.5(3/4-2/3)-0.5(2/3-2/4)

The problem with your logic is that when you have two heads out of three, if you go again and get tails you won't stop there, so the "2/4" is not correct. Having said that, I don't know what the correct answer is.

Bunny Buster

Sounds like a bit of a " how long is a piece of string " one to me!!:D

Fremen

There are some related comments here. Trouble is, the problem is phrased in a slightly different way and all the answers look right to me, even though they claim different results.

http://mathoverflow.net/questions/17960/google-question-in-a-country-in-which-people-only-want-boys

RoundTower

I didn't read all the answers to that one, but I'm certain the answer in that one is 50-50. Every child's sex is independent to the sex of all the other children. Note that it asks a very different question to the OP here.

MoogPoo

I love these, my intuition is always completely wrong for stats questions.
Here's another one: if your checking the number of males/females in houses, and at a particular house there are 6 children and you know that 5 are girls. Whats the probability that the 6th child is a girl?