Game Development Reference
In-Depth Information
On the first flip, we have a 50% chance of winning the $1 pot. So our expected
winnings after one round (
E
1
) are
We can expect to win an average of 50 cents on the first round. If we were to be
betting
only
on one flip of a coin, we would be done now. We can expect to win 50
cents on average, so we would be willing to bet 50 cents on the game. In theory, over
time, we would hope to break even with this. If we are allowed to wager less than 50
cents, that's a bonus, as we would most likely come out ahead in the long run. A
wager over 50 cents is not in our best interests, as we are going to be bleeding cash
the longer we play. (Although that doesn't seem to stop people from lining up at
casinos, does it?)
Our little lottery game doesn't stop there. If we pass the first flip (50% chance),
we then have another 50% chance of winning $2. Therefore,
Third flip?
Or, to simplify,
So now we can expect to win $1.50 on average after three flips. It seems that, if
we were to stop here, we should wager $1.50 because we have a chance of winning
that after three flips.
The peculiarity comes into play when we extend things out to infinity. After all,
in theory, we could flip heads an infinite number of times in a row before throwing
that tail and collecting our winnings. Looking at it in a formula,
