The question you opened
The 200 dollar bonus with strings
A 200 dollar bonus needs 20 times the bonus wagered on a game that returns 95 cents per dollar.
About how much of the bonus survives the wagering, in dollars?
$0
Twenty times 200 is 4,000 wagered, and 5 percent of that is the whole 200.
The bonus and the condition cancel exactly. A wagering requirement of 20 times 200 dollars means 4,000 dollars has to pass through the game before anything can be withdrawn. A game returning 95 cents per dollar keeps 5 cents, so the expected cost of pushing 4,000 dollars through it is 5 percent of 4,000, which is 200 dollars. That is the bonus, to the cent.
The general formula is short. The expected value of a bonus is the bonus minus the house edge times the wagering requirement times the bonus. With an edge of e and a multiplier of w, the bonus is worth B times 1 minus e times w. It breaks even when w equals 1 over e. At a 5 percent edge, break even is a 20 times requirement.
Anything above 20 has negative expected value, and requirements of 30 or 40 times are common. Two details push it further against the player. Many bonuses exclude the low edge games, so the effective edge is higher than the one advertised on the headline game. Many also cap the amount that can be withdrawn from bonus winnings, which truncates the good outcomes while leaving the bad ones intact.
Variance is the one thing on the player's side. Expected value of zero does not mean every outcome is zero. Some players will finish ahead and some behind, and the spread is wide. That is what makes the offer feel like a gift to whoever gets lucky and gets talked about.
The habit worth carrying out of this card is to multiply any small percentage by the volume it applies to before deciding it is small. Fees on trades, spreads on currency, and management charges on investments all work this way: the rate looks negligible and the turnover is what makes it real.
Technique: Edge times turnover