[PollyEmory]

I'd like to give this a shot if no one minds. I'm new though, so be kind.

pzhon is right in that every single event of the flip is -EV for the player. The same way any single hand of blackjack is -EV for a player unless there is something like a bonus to offset the expected loss. The limited bankroll is that bonus.

If you think of each coin flip in the progression of coinflips as a decision tree and assume equal bankrolls in either direction, the player will bust in more then 50% of the eventual outcomes. However (!!!) because of the house limited roll, some of the tree will be artifically pruned out. How much depends on how small the casino bankroll is with respect to the bet size and not with respect to the player's bankroll. If the casino has enough BR to cover three losing flips before busting and the number of flips needed to approach EV is 100, it doesn't really matter if the player has BR of 15k or 150k.

Let's see if I can demonstrate my point.

$50 Flips

1st Flip(Casino = 100, Player = 1500)
a. Player won (Casino = 51, Player = 1549)
b. Player lost (Casino = 150, Player = 1450)
a.1 Player won(casino = 2(basically broke), Player = 1598)

Now if we could continue this tree indef. downwards, the number of leaves where the casino makes money would exceed the number of leaves when the player makes money. However, the limited casino bankroll means that the tree will be artifically pruned once the casino goes broke, therefore reducing its edge. By how much is a calculation I can't really undertake at 6am est, but offhand, I'd say it depends on casino BR v betsize. The higher the ratio, the less edge it gives up.

So yes, the any single event of the flip is -EV if EV is computed based on infinite number of tosses and infinite player/casino bankrolls. In the real world, casinos' rolls are basically an approximation of infinity. But if the roll is small, the approximation is bad. That skews results. Therefore situationally, this could be a +EV gamble for the player. I'm sure someone will step up and either back me up with some rigorous analysis or disprove me with the same. I suspect if you plot expectation vs finite bankrolls of various sizes(BR) approximating infinity (INF) in EV calculations, you will find that under certain values for BR (low v betsize), the EV will be positive for the player.

In other words, if we played out this situations infinite amt of times (casino BR $200, bet size $50, player BR $1500) the player will bust the casino more times then casino will make money off this deal. But the higher the BR, the more precise the approximation, the closer actual expectation will approach theoretical expectation.


--Polly