Update: Understanding the Terms and conditions is key when evaluating promotional offers. It turns out that this particular promotion has some hidden caveats that lessen the overall EV. Skip to the Updated Answer below for details. If the "first bet insurance" was a refund of your bet into your account, which is what it sounds like until you read the fine print, then the original answer would apply.
Original Answer:
Your math lands on the theoretical correct answer, though in practice a casino won't payout evenly on a 50/50 chance, so there needs to be an adjustment.
A "first bet insurance" promotion (typically for betting on outcomes of professional sports) means only your first bet is insured, so let's assume you max that at $1100. A true 50/50 chance would normally have odds of -110, meaning you bet $110 to win $100, so your $1100 bet would have a 1/2 probability of winning you $1000. (This is how casinos make their profit .) If you lose that first bet you'll be given the $1100 back, and you typically have a certain amount of time to wager that amount again, after which time you'd be able to withdraw any winnings. Usually you aren't required to wager that second $1100 all in one shot, but for simplicity let's assume that you do bet it all on another 1/2 probability bet with a -110 line. Your chances of losing twice = 1/2 * 1/2 = 1/4, and therefore your chances of winning are 1 - 1/4 = 3/4.
The Expected Value (EV) would be:
EV = (1000 * 0.75) + (-1100 * 0.25) = $475
The $475 positive value certainly makes this gamble worth considering, however, the most important thing to realize is that your actual outcome is going to be either winning $1000 or losing $1100. As with all gambling, regardless of how good the "deal" is, you must be OK with the possible outcome of losing $1100. If you would lose sleep over the worst outcome, then you should either lower your bet to where you won't care about losing it, or not do it at all.
Can you do better than an EV of $475? Let's hedge!
Conceptually, either you are going to win a $1000 or the casino is essentially giving you $1100 for free to gamble with. Ideally you would just walk away with that $1100. Can we figure out a better way to keep (most of) it?
In general, online casinos have a built in profit margin that makes it impossible to profitably hedge your bet. For example, suppose both teams have an equal chance of winning a sports game. If you bet $110 on each team winning with odds of -110, you would lose $110 and win $100, for a loss of $10 or approximately 9% of your wager.
However, promotions open the door to profitable hedging techniques. For example, suppose you deposit $2200, and find an even up bet and you wager $1100 on both sides of it. One side will win you $1000, and the other side will lose with a reimbursement of $1100. Then you split the reimbursement and do the same thing on another bet of $550 each. One side will win $500 and the other loses $550. Your total winnings would be:
-2200 (initial buy-in)
+2100 (first winning bet 1100+1000)
+1100 (first losing bet reimbursed)
-1100 (second bets of 550 each)
+1050 (second winning bet 550+500)
-----
+ 950 (total winnings)
In this case the EV of $950 is also the guaranteed winnings!
It's very likely that the terms and conditions of the promotion will not allow you to hedge in this way, and would probably void the insurance if you do. That being said, it would be extremely easy for them to detect this if you do it yourself, and extremely hard for them to detect this if you did it with a friend. If you do it with a friend your second set of bets would each need to be $1100 instead of $550, so the last two lines of the calculation would be -2200 and +2100 respectively, reducing the total winnings from $950 down to $900. The reason this is slightly lower than without the partner is because you are making larger bets, one of which is guaranteed to lose.
Updated Answer:
Unfortunately, the particular promotion in question has some caveats that lessen the overall EV. Specifically:
- Awarded Free Bet is a one-use token and cannot be broken down across multiple bets.
- The value of the Free Bet is not included in any winnings paid out.
Rule #11 is a bummer because if you win the second time, you only keep your winnings. Continuing from the previous calculations, the second winning bet would pay $1000, but without retaining the original $1100 bet it becomes a loss of $100. The updated EV calculation would then be:
EV = (1000 * 0.5) + (-100 *0.25) + (-1100 * 0.25) = $200
Rule #11 also means you cannot guarantee a win even with the hedge. Note the bet token makes the second half of the hedge pointless because not only can't you split it, but even with a partner your choices are to either win $1000 without a hedge, or pay $1100 to win $1000 with a hedge, so you should just skip the second hedge bet altogether:
-2200 (initial buy-in)
+2100 (first winning bet 1100+1000)
+ 0 (first losing bet reimbursed)
- 0 (second bet token)
+1000 or 0 (second bet wins or loses)
-----
+ 900 or -100 (total winnings)
So with the hedge your EV becomes:
EV = (1000 * 0.5) + (900 *0.25) + (-100 * 0.25) = $700
This is still a great EV, but there's also a 25% chance you'll lose $100.
Disclaimer: I'm definitely not endorsing or condoning any of the ideas in this answer; I'm simply stating the math behind them.
