Arbitrage: when both sides cost less than the payout
Every price is a cost for a payout. A bet at +150 costs 40¢ for every $1 it pays back; a prediction-market contract at 55¢ plus a 2¢ fee costs 57¢ for $1. One place usually prices both sides of a game so the two costs add up to more than $1. That extra is the vig, and it is how the house is paid.
Different places do not move together. When one book is slow to follow a move, or an exchange's traders see a game differently, the cheapest price on one side and the cheapest price on the other can add up to less than $1. Back both, in the right amounts, and you are paid the same whichever side wins. That is an arbitrage.
Splitting the stake
Stake each side in proportion to its cost. With costs of 57¢ and 40¢ per $1, a $97 stake buys $100 of payout on each side: $57 on the first and $40 on the second. Whichever wins pays $100, a $3 profit on $97, about 3.1%. Round the stakes to whole dollars and the two payouts drift apart slightly, so the profit is the smaller of the two.
The terms must match exactly
Two bets only cancel out if they settle on the same result. A sportsbook moneyline that counts overtime does not cancel a contract that settles on regulation time; a spread of 3 can push while a spread of 3.5 cannot. If the terms differ, there may be a result where both bets lose. A real arbitrage needs the same game, the same period, the same line and no way for one side to push while the other loses.
Postponements are the one difference that can remain. A sportsbook usually voids a postponed game, while a prediction market may keep the contract open or settle it at a fair price. Check how each side handles a postponement before you place both legs.
What it costs you
Prices move in seconds, and a book can refuse or reduce a stake. Place the leg most likely to move first, and only when both prices are still there. An exchange only fills as much as its order book holds, so a large stake can fill at worse prices than the first one shown.
Sportsbooks also watch for arbitrage. Accounts that only ever take the sharp side of a mismatch often get their limits cut. Many bettors treat arbitrage as a short-lived edge, not a way to bet for years.
Example. (illustrative prices, not live data) One book has the Bills +3.5 at +105; an exchange has the other side, Chiefs −3.5, at 47¢ plus a 1¢ fee. Costs: 100 ÷ 205 = 48.8¢ and 48¢. Together 96.8¢ per $1, so $96.80 split $48.80 and $48.00 pays $100 either way: $3.20, about 3.3%. Both are half-point lines, so neither can push.