What Is the Fair Price per Trial When Stopping After 3 Heads?

[zli034]

Say,

I'm a bookkeeper of a gamble of flip coin. The price for each trial is 0.5, i.e. if there is a head I pay gambler 0.5, otherwise I get 0.5 from the gambler. There are only 10 flips or trials in the game, so that each gamble only can play 10 trials. I know to choose the 0.5 as the fair price because, I'm not going to lose money on average.

But if I modify the game, and do not fix the number of trials, only stop the game sequence when the 3rd head appears. What is the fair price for each trial so that I don't lose money.


Thanks

 

[chiro]

Hey zli034.

If all tosses are independent, then you should look at the negative binomial distribution, its expectation, and its variance:

http://en.wikipedia.org/wiki/Negative_binomial_distribution

 

[ssd]

In the 1st case, the statement "price of each trial" is meaningless. The idea is, for a fair (!) coin, payment by gambler and you should be same.
In the 2nd case expected number of tails before the game ends is 3 (for a fair coin).