Why 52.4% is the most important percentage in sports gambling.
By: John Culver
Tags: NFL, Sports, Las Vegas, NFL Picks, Sports Betting

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Albert Einstein putatively claimed that compound interest was the most powerful force in the universe, declaring that: “He who understands it, earns it…He who doesn’t…pays for it.”

The simple mathematical logic supporting compound interest’s power to exponentially grow capital over time is irrefutable. Anyone in finance, or with a cursory understanding of bean-counting, knows that time is money. In a similar vain, the fact that sports gamblers must win or cover bets over at least 52.4% of the time is an inevitable hurdle all sports gamblers must clear to be profitable.

What’s the deal with 52.4%?

The economics of sports gambling is constructed so that the house or sportsbooks typically require players to wager at least 1.10x to win x. Put another way, most bookmakers give 10 to 11 odds — forcing the gambler to risk $11 to win $10 contingent on the outcome of a binary event (e.g., win or lose, cover the spread or not). If your team wins or covers, you win $10 (in addition to your $11 depository wager); if not, the bookmaker keeps your $11. And if the outcome is a push (e.g. the margin of victory = the spread), no money is exchanged.

To better illustrate this concept and explain the derivation of 52.4%, let’s turn to an expected value formula [see Ex. 1.0, below].

Ex. 1.0: How Vegas Works

Sportsbooks profit by either: being on the winning side of wager and/or; through earning risk-less profits via structured commissions embedded into the gambling odds.

Let the expected value (EV) of a wager be defined as:

EV = W(p) — L(1-p)

W = potential cash winnings ($)

= potential cash winnings ($) p = probability of winning (%)

= probability of winning (%) L = potential loss or amount risked to win W ($)

= potential loss or amount risked to win W ($) 1-p = probability of losing (%)

You risk $11 to win $10 if Team XYZ covers the spread.

$10p — $11(1-p) = 0

To solve for p, set the expected value or expected profit equal to zero.

p = $11 / ($11+$10)

p = $11 / $21

p = 0.5238 or apx. 52.4%

As this example demonstrates, p = 0.524 is the value that makes our expected profit equal to zero. Therefore we can conclude that 52.4% is the frequency or rate that we must clear to make a profit, or wager to earn expected values > 0.

If this embedded edge did not exist, bettors would merely need to win 50.1% of the time to be profitable.

How Vegas Works: Bookies’ Indifference

We can expand on the previous formula by incorporating the profit opportunity derived from bettors’ preference of wagering on a favorite:

E(Bookmaker Profit) = [(1-p)f + p(1-f)]* (1+v) — [(1-p)(1-f) + pf]**

p = probability that a favorite wins a particular game

= probability that a favorite wins a particular game f = fraction of total currency wagered on the favorite out of total currency wagered on the game

= fraction of total currency wagered on the favorite out of total currency wagered on the game v = vigorish (vig) or commission levied by sportsbooks, which is paid only on losing bets.

= vigorish (vig) or commission levied by sportsbooks, which is paid only on losing bets. *fraction of currency bet in which the sportsbook wins [(1-p)f + p(1-f)]. This quantity is multiplied by (1+V) to reflect the commission.

**payout to bettors in the event that the bookmaker loses.

We can simplify the previous equation further by rearranging the terms to:

E(Bookmaker Profit) = (2 + v)(f + p — 2pf) — 1

p = probability that a favorite wins a particular game

= probability that a favorite wins a particular game f = fraction of total currency wagered on the favorite out of total currency wagered on the game

= fraction of total currency wagered on the favorite out of total currency wagered on the game v = vig or commission levied by sportsbooks, which is paid only on losing bets.

According to the Freakeconomics economist, Steven Levitt, “If either the probability the two teams win is equal (p = 0.5) or the money bet on both teams is equal (f = 0.5), the bookmakers gross profit simplifies to v/2. In either of these instances, the bookmaker is indifferent about the outcome of the game and earns a riskless [sic] profit proportional to the size of the commission charged.”

To ensure profit, the bookmaker need not be able to predict the outcome of games more accurately than bettors — the bookmaker merely needs to predict bettor preferences so as to balance wagers on each side.

Given the aforementioned, let’s solve for the expected value (EV) of a fair coin toss from a bettor’s perspective utilizing the following expected value equation (below):

EV = W(p) — L(1-p)

Should you take this wager?

EXAMPLE 1.1

You risk $11 to win $10 if the fair coin flip comes up heads.

$10p — $11(1-p) = Expected Value (EV)

p = .50 = the probability of winning a bet (heads).

1-p = .50 = the probability of losing a bet (tails).

$10 = The amount awarded to the gambler if heads.

$11 = The amount risked by the gambler to win $10, which goes to the house if the gambler loses (tails).

(10* 0.50) — (11* 0.50) = -0.50 = Expected Value (EV)

To calculate your expected long-term return on investment (ROI), simply divide the expected value by the amount risked:

[(EV) / (Amount Risked)] = ROI

(-0.50/11) = -4.55%

Now Imagine that instead of making a spread bet, you were betting on a fair coin toss with 50–50 odds (i.e., 50% win, 50% lose).

A: You should never take this wager as your expected value is negative [EV < 0]. Consequently, your expected long-term return on investment (if you were to keep playing this game with the same odds over time) would reduce your bankroll by -4.55%.

Now imagine a scenario with a rigged coin whereby heads has a 52.4% chance of being flipped (p = 0.524) and tails has a 47.6% chance being flipped (1-p = 0.476) then your expected value would equal zero (as demonstrated in Ex. 1.0). While you won’t lose money in the long-run playing this game, you certainly won’t make any!

How Vegas Works: Break-Even?

*EV = gambler’s expected value; **BEP = gambler’s break-even point or percentage of bets needed to win to break-even.

An efficient way to break the bank…

The Vigorish: How Sportsbooks Collect Risk-less Profits

In traditional bookmaking parlance, the vigorish, often disparagingly referred to colloquially as the vig, juice, or rake is the theoretical risk-less average profit per unit collected by a sportsbook assuming a roughly even split of betting currency on each side (e.g. 50% of wagers on the favorite; 50% on the dog).

Returning to Ex. 1.0, let’s assume that 50% of the betting action supported Team XYZ covering the spread whereby Team XYZ backers risked $11 to win $10 per bet in the event that Team XYZ covers. Furthermore, let’s assume that the sportsbooks collected an equal proportion of wagers supporting Team XYZ’s opponent, Team ABC covering the spread whereby ABC spread bettors also risked $11 to win $10 per bet.

So long as the game’s final margin does not land on (push) the spread value, sportsbooks will earn a guaranteed profit of $1 per bet [see Ex. 1.2 below for further explanation].

EXAMPLE 1.2

The house has collected even money on the spread of +/-5.5 between Team XYZ, the favorite (-5.5), and Team ABC, the underdog (+5.5).

50% of the money bet on Team XYZ to cover -5.5

50% of the money bet on Team ABC to cover +5.5

Bettors for each side risked $11 to win $10.

Game Outcome: Team ABC covers the spread +5.5; Team XYZ backers lose and forfeit their $11 per bet.

The house pays out the Team ABC supporters at $10 per ticket, and holds the $11 per Team XYZ wager, retaining a guaranteed profit of $11-$10 or $1.

To recap:

Team XYZ bettors risked: $11 Team ABC bettors risked: $11 The house collected: $11 + $11 = $22 (prior to the game)

The house retained $1/$22 or 0.045 (4.5%) as the vig, returning the $11 risked plus the $10 in winnings to the Team ABC backers.

How to conceptualize probability and hurdle rates

Expected Value [EV] / Amount Risked $ = Return on investment [ROI %]

The ROI is best conceptualized as your expected return per wager in the long run. Our human minds are not predisposed to naturally think in terms of probability, incidentally we tend to incorrectly interpret a loss for a given event with a 52.9% probability of occurring (47.1% probability of not occurring) as having a 0% of occurring because it did not go our way.

It is helpful to keep a quarter handy to anchor the mind to what 50% feels like as a heuristic to better calibrate our schema in terms of chance. Flip the quarter and call it 10 times, and tally how many times you called it properly. Remember that a 52.9% likelihood is not very different (in terms of probability) from 50%!

Like compounding, intelligent wagering is a long-term game. All of these hypothetical values are long-term returns that ought to be achieved with time and a sufficiently large sample size assuming your fundamental odds are more accurate than the market, and that you only wager when the EV > 0.

Expected Value Table

Risk $110.00 to Win $100

Plug when p(w) > 52.4+n [n = +0.005 increments]

p(w) = 52.9%

EV = (0.529 * $100)-(0.471 * $110) = 1.05, ROI = 1.05/$110 = 0.95%

***

May the odds be forever in your favor.

***

Works Cited

Levitt, Steven D., “Why Are Gambling Markets Organized So Differently From Financial Markets?” The Economical Journal, 114 (April), 223–246. Royal Economic Society 2004. Blackwell Publishing, 9600 Garsington Road, Oxford, UK, p. 227.