As part of a casino promo, I got a $100 free bet to place on any table game. It had to be all used in one bet. I played a hand of blackjack, lost, and moved on.

I started thinking about it later, though, and I realized that the EV of playing a hand of blackjack in that way is about $48-$49, I believe. (Is that about right? You'll win $100 a bit less than 50% of the time, right?)

Then I started assuming that all bets must be about the same EV... the $100 free bet would always be worth a bit less than $50 due to the house vig.

But wait: if I put the $100 all on a single number in roulette, I could win $3500 (not $3600, since you don't get the money from the bet itself) 1 in 38 times. That works out to an EV of over $92.

Is that possible, or am I doing something wrong with my math? Did I really throw away $43+ of EV by playing blackjack instead of betting a single number in roulette? And how could the difference be so vast? Finally, is there anything even better than the roulette option that I could choose the next time I'm in this situation?

Here's why the difference is so vast. Consider a game that is completely and simply balanced to give neutral EV. Like, say the game is "choose a number between 1 and N" where it pays out n-1 times your bet when it hits.

So let's say you're betting $100
When you lose, you lose $100. This happens (n-1)/n times
When you win, you win n-1, this happens 1/n times
EV = -100*(n-1)/n + 100*(n-1)/n
it should be clear that this is 0EV.

However, when you are fronted the $100, the first argument just disappears and you are left with
EV = 100*(n-1)/n

So, if n=2, you'll win half the time and your EV = 100/2 = 50
If n=100, you'll win 1/100 times and your EV = 100*99/100 = 99
and so forth.

No hal, your calculation is wrong. There is no "-100" as this is a free bet.

Blackjack: EV = 0.49*100 = 49
US Roulette: EV = 1/38*3500 = 92

OP - you are correct - it is much better EV to place a FREE bet on a high odds proposition. That is why most b&m casino free bets cannot be placed on a single number in roulette, only an even money proposition.

FaceSlap

But, thinking about it a bit more, while the EV is much greater for roulette, your comparing a (roughly) 3% win probability vs a (roughly) 50% win probability with obviously different payouts.

I guess it depends on your utility-risk posture.

Absolutely.

Assuming you can access any layout bet you do have a lot of choices. E.g. on a standard European roulette table you have 1/2/3/4/6/12/18 out of 37.

I suppose it depends what 100 and 3500 mean to you. If it was a million dollar free bet I'd certainly put in on red. If it was a ten dollar free bet I'd certainly put it on 29. If it was 2k... hmmm.



POP QUIZ: Which two free roulette bets with different odds have exactly the same EV?

Thanks, you guys. I'll have to do some digging as to the specific rules next time. I'd be willing to throw the $100 on a longer shot next time if they allow it.

Tthink of the difference between giving you a real dollar with the stipulation that you bet it and a free dollar bet that is taken away win or lose. Getting high odds its almost identical because you almost always lose. At even money the difference between the two "gifts" EV wise is 50 cents.