## Interesting gambling theory question/puzzle

I saw a discussion the other day that reminded me of a problem I came up with and I posed it to a bunch of friends. None of them were able to get the right answer and reasoning, so I figured it would make a good blog post.

I originally thought of this in the old days when online casino bonuses were prevalent. If you’re not familiar with them, you would deposit some amount, get a bonus, and have to meet a wagering requirement to withdraw the bonus. So you’d do something like deposit $100, they’d give you $20 on top of that for free, and you’d need to wager $1000 at blackjack to withdraw. Blackjack pays back around 99.5% if you play perfectly, so you’d end up losing $5, thus profiting $15 after the bonus. This is actually how I started my bankroll many years ago.

So, here’s the problem:

You deposit $400 in an online casino and are given a $100 bonus immediately, so you have $500 to bet with. You can withdraw only after betting a total of $2500. Let’s say you play a game where you flip coins and if you win you get 1.99 times your bet and nothing if you lose. This has a 99.5% return like blackjack, but I’m abstracting it because in blackjack you can run into bad EV spots where you make a bet and then don’t have enough to split or double down. The table limits for this game are minimum bet $1 and maximum bet $100. How much should we bet to maximize EV, and **why**? Does it even matter? If so, why? If not, why not?

Time and variance are not factors in this problem. We don’t care that making lots of small bets takes more time than making a few large bets and we only care about maximizing EV.

Post answers or discussion in the comments. I’ll reveal the answer in less than a week and may drop some hints if nobody gets it.

## Doug Shultz said:

why start a -EV bet? But the 100$ bonus makes it a positive EV.

Go ahead, it really doesn’t matter in a coin flip scenario since you’ll win 50% of the flips over time. You should maximize the number of individual wagers so that you’ll tend towards winning 1/2 the time.

Bet a buck each and after 2500 wagers you’ll end up with $487.50 paid to you. Betting more on each individual bet will probably skew your away from the 50/50 winning percentage of this bet.

## Aaron Crawford said:

Assuming that:

1. You want to stop playing as soon as you clear the bonus because the game isn’t getting any less -EV, which means betting exactly $2500.

2. You’re OCD and have to bet the same amount every time.

The only rule I can come up with is to pick a factor of 2500 so you don’t end up having to bet $3 up to $2502 and losing a few extra cents. Other than that, it looks like like you end up with $487.50 whatever bet amount you choose.

It’s possible I’m overlooking something, since that seems like a pretty obvious answer.

## Derk’s blog » Blog Archive » Theory problem update said:

[...] been a few days since I first posted my puzzle. If you missed it, you can find it here. Aside from the comments there, there is also some discussion about it going on over at Poker VT [...]

## Derk’s blog » Blog Archive » Gambling theory problem/puzzle solution said:

[...] originally posted a problem about a week ago here and an update with a hint here. If you haven’t read those, please do and give it a try before [...]