I am researching about betting arbitrage and I always like to understand why formulas solve for certain things when based on their variables in the equation, don't seem like they would solve for said thing. In this case, I am wondering about a formula for betting arbitrage. I understand how to determine the market-derived implied probability (including a vig) of an outcome with fractional odds or decimal odds and the intuition behind it.
What I cannot understand is the formula of how to determine how much to bet on each outcome to guarantee yourself arbitrage. (given an event where the implied probability of both possible outcomes sums to less than 100%).
Given a game between 2 teams where their odds provide implied probabilities under 100% (some websites also call this the arbitrage percentage), there is an arbitrage opportunity available.
Can someone please explain to me the intuition of how the formula of how much to bet on each team to obtain the available arbitrage makes sense given its variables? Formula below:
Stake for Team A = (Total Stake * Team A Arbitrage Percentage)/Total Arbitrage Percentage
Suppose a certain sports event has 3 possible outcomes: Team A wins, Team B wins, or draw. You find the best available odds for betting on each of those outcomes. The intuition is: bet an amount on Team A that guarantees you'll win $\$100$ if Team A wins. Then bet on team B enough so that you'll win $\$100$ if team B wins. And same for draw.
Now you've placed 3 bets and you will win $\$100$ no matter how the sports results goes. But how much money have you spent placing those bets? If the total is $\ge \$100$ then you've broken even or lost money; this is not an arbitrage opportunity. If the total is $< \$100$ then you're guaranteed to make money, since you'll definitely win $\$100$ back.
The formulas on your website just come down to calculating a) how much money you need to bet on each outcome, and then b) deciding whether you'll make or lose money if you try it.
Note in the example above I assumed there are 3 outcomes, but in real life you would bet on ALL possible outcomes. For many sports matches there would be only 2 outcomes, but for some events there could be 4+ possible results, and you'd need to bet on each of them. Once again, the core intuition is to bet on all possible results so that you'll be guaranteed to win exactly $\$100$ no matter what, and then if you can pay $< \$100$ to place that combination of bets then it's an arbitrage.