Understanding the Probability and Expected Value of Parlays

There are a lot of ways public bettors hand over hard-earned money to the house while giving up more of an edge to the book than they should.  Understanding payouts and combined win probability allows bettors to have a true understanding of their expected value when placing a wager.  Let’s start with the simple math of a 1-game straight bet with the standard price of -110.

If a bettor makes two separate wagers for $110 to win $100, wins one and loses one, they would have a return of -$10 while owning a 50% win rate.  For this reason, a bettor must have a win percentage greater than 50% in order to have a positive expected value when placing a bet.  Extrapolating on this math to make the numbers simple, if a bettor loses 100 bets of this size, they will have total loses of -$11,000.  In order to breakeven, that bettor must win 110 bets to have $11,000, for a total profit of exactly $0.  110 wins out of 210 total bets (110 winners + 100 losers) is a win percentage of 52.38%.  Therefore, in the long run, if a bettor wins over 52.38% of their bets (assuming -110 action) they will make a profit, win less than 52.38% and they are destined to lose money. 

This is not only simple math, but also makes it easy to make decisions when placing a bet.  If they have a win expectancy over 52.38%, place the bet and collect cash over the long-run.  If it’s a coin flip and only a 50/50 proposition, don’t place the bet unless losing money sounds like a good time.  Things get slightly more complicated when going from a single wager to a parlay.

Two-game parlay payouts will vary slightly from book to book, but in general, they are 13/5 for two-games that are priced at -110.  The payout of 13/5 equals 2.6 to 1 e.g. win $260 for a $100 wager.  The payout of 2.6 to 1 equals an implied win percentage of 27.78%.  Carrying over our previous math, winning 38.46 times would produce $10,000 in winnings (38.46 x odds of 2.6 x $100).  The breakeven math is 100 losses at $100 per wager for total losses in $10,000, resulting in $0 gained.  Therefore, 38.46 wins divided by 138.46 total wagers equals winning on 27.78% of wagers placed.

Only needing to win 27.78% of the time to break-even sounds a lot better than needing to win 52.38% of wagers placed, but let’s not forget that we need to win two games to cash.  Let’s assume a 50% win percentage for a bettor who does not have a proven system.  There are two games with a 50/50 chance of hitting which results in a combined win probability of 25% (0.5 x 0.5 = 0.25). 

…Whew, now that we have the math out of the way, let’s compare the profile of a two-game parlay vs. a straight wager.

These two examples illustrate exactly how and why, in the long-run, public bettors lose money to the books.  Paying juice, whether it is in a straight bet or baked into the price of a parlay, tilts the scales in favor of the house.  The only way to successfully make money in the long-run is to have a win percentage greater than 52.38%, but also understanding what the combined probability and implied win percentage is whenever making a parlay or teaser bet.

The combinations of payouts, probabilities, and expected values are endless when considering the factors that need to be accounted for:

Number of legs in the bet

    • we’ve discussed 1 and 2-games, but most books allow up to at least 8 games in a parlay, if not more.  The combined probability of a win, as well as the amount of vig baked into the payout all need to be calculated differently depending on the number of legs in the bet.

Price/Odds

    • the assumption used for the previous examples were -110 action, but as we all know, the price for each individual game can vary by the second.  The juice is a formulaic, but important, factor when determining the overall payout and in turn, the expected value associated with placing the wager.

Win Expectancy

    • this is the most critical, and often, least understood part of the equation.  The average bettor wins 49% to 51%, as evidenced by historical data from the Las Vegas Super Contest.  Most gamblers like to talk about how much money they have won (or lost), but in reality, with good money management, the win percentage is going to dictate the long-run returns.

We at ProMathletics understand a) most people place bets with the hopes of making a return b) most people don’t have the knowhow and/or time to calculate the expected value for each wager placed.  There are a lot of Parlay Payout Calculators that simply tell the user what the payout will be if all the bets hit.  That’s ok, but that functionality can be achieved on most betting apps...what truly matters is not the payout, but that the bet actually has a positive expected value when it’s placed!  Fortunately for you (and us!), all of our win percentages have historically been at profitable levels.  Check out our moderately priced subscriptions to receive our proprietary research and start winning today!

Learn more about our historical win rates and the long term value of betting parlays with positive expected value.