What Are the Odds?
Copyright © 2023 by Stan Brown, BrownMath.com
Copyright © 2023 by Stan Brown, BrownMath.com
Summary: You can say the probability is 25%, or you can say the chances are one in four, or you can say the odds against are three to one. Those all mean the same thing, but how to find equivalent statements in real life isn’t always obvious. This article explains how odds and probability are connected, and gives you easy formulas for converting one to another.
In statistics class, things are simple. Probabilities are always a decimal 0 to 1 or a percentage 0 a 100%, which of course is the same thing. In real life, there’s no one standard way to make probability statements, so you have to look at each statement carefully and interpret it.
What does “a chance in a thousand” mean in probability terms? It says a given event will happen about one time per thousand trials, so p = 1/1000 = 0.001 = 0.1%.
“Four chances in ten thousand”? p = 4/10,000 = 0.0004 = 0.04%.
“Two chances in nine”? p = 2/9 ≈ 0.2222 ≈ 22%.
This is so straightforward it doesn’t need a formula — just divide the first number by the second and, if you want to, convert to a percentage.
Sometimes you’ll see something like “only one chance out of 50”. In this situation, “out of” is mathematically the same as “in”, so again you just divide: p = 1/50 = 0.02 = 2%.
What about going the other way, converting probability to chances? Here you convert a decimal probability to a fraction, and there are always multiple answers. 1 chance in 2 is the same probability as 2 chances in 4, 11 chances in 22, 500 chances in 1000, and so on.
You get to pick either the first number or the second number — not both, of course. While mathematically either one could be a decimal, for people to understand them you almost always want whole numbers. Here are some suggestions:
For example: An automaker has warranty-eligible defects in 6.8% of its cars. That’s about 1/14.7. You could say “a little more than 1 in 15”, but let’s look at some options: 6.8% = 6.8/100 = 68/1000 = 3.4/50 = 1.7/25. You don’t want a decimal in your statement — what would “6.8 chances” even mean? So “68 chances in 1000” is probably your best bet, or you might approximate with “just under 7 chances in 100”.
For instance, if you grab one M&M at random from a bag, let’s suppose there’s a 24% probability it’s blue, 20% orange, and 13% brown. (I’m ignoring the other colors.) 24% = 24/100, 12/50, or 6/25. There are other fractions, an infinite number in fact, but those are the simplest ones. 20% orange is 20/100, 10/50, or 1/5. But for brown, 13% = 13/100, and you can’t get smaller numbers if you want whole numbers in top and bottom. So the natural thing is to use the fractions with 100 on the bottom, and say you have 24 chances in 100 of drawing a blue, 20 in 100 for orange, and 13 in 100 for brown.
Odds are kind of hard to get your head around because there are so many different kinds. To start with, “odds” as statisticians and scientists use the term are different from “odds” as bookmakers and bettors use it. Odds in science and math could be called “pure” or “actual” because they come from a straightforward computation from probability, but in betting odds are typically skewed to provide a profit for the bookmaker.
Odds are spoken as “number to number” and written number:number, number–number, or number to number. Unless it’s very clear from context, you have to say whether you’re giving odds for or against.
The science-math crowd and the betting crowd tend to approach odds differently. It’s not 100%, but the general trend of differences is shown in this table:
|In Science and Math||In Betting|
|Odds are||One ratio, fraction, or decimal||Two whole numbers|
|Commonest punctuation||Colon :||Dash –|
|Preferred type||Odds for, odds of, odds in favor||Odds against, odds on|
|Match to probability||Pure formula||“Fudged” to make a profit|
We’ll start with the science-math uses, then move on to odds in betting.
The odds for an event, of an event, or in favor of an event are OF = p/(1−p), and the odds against the event are OA = (1−p)/p. You can see that OA = 1/OF and OF = 1/OA.
Let’s say that the probability of an event is p, for instance a 20% chance of precipitation tomorrow. The probability it won’t happen is 1 − p, 80%.
With p = 20% or 0.2, the odds against precipitation tomorrow are 0.8/0.2 = 4/1 = 4:1. The odds for precipitation tomorrow are 0.2/0.8 = 1/4 = 1:4.
You already know that OF = p/(1−p), so all you have to do is solve for p:
p/(1−p) = OF
p = OF − p×OF
p + p×OF = OF
p(1 + OF) = OF
p = OF/(1+OF)
(If your odds are expressed with : or –, replace that character with / to make a fraction.)
If the odds in favor of rain tomorrow are 1:4 = 1/4 or 0.25, the probability of rain tomorrow if p = 0.25/(1+0.25) = 0.20 or 20%, exactly what we had at the beginning of the example.
If you need probability of non-occurrence, subtract from 1: q = 1 − 0.20 = 0.80 or 80%.
What if you have 4:1 odds against rather than odds for? Remember that OF = 1/OA, so take OF = 1/(4:1) = 0.25 and substitute in the formula above.
What do you make of a statement like “bookmakers are quoting odds of 3–1 for Freebiscuit to win”? (The dash is pronounced “to”, so the odds are “three to one”. Some people use a colon, 3:1, also pronounced “three to one”. You’ll sometimes hear “three to one against”, but usually the “against” is just understood.)
One thing tu understand is that betting odds aren’t just about probabilities. They do relate to the bookmaker’s guesses at probabilities, but they’re also skewed to provide a profit for the bookmaker. In “Horse Racing Odds Explained”, TVG says:
Horse racing odds are the return you can expect on your investment if your wager is successful, the odds or the payout reflecting to some degree the percentage chance your horse has of winning the race. The lower the chances of success, the bigger the payoff will be.
For example, if you place a $1 bet at 3–1 odds and the horse wins, you get your original bet of $1 plus a $3 profit, totaling $4. Odds of P to B, P–B, or P:B tell you that if you bet B dollars and win, your profit will be P dollars. (It’s profit because you also get back the $B that you bet.) For a $2 bet, just multiply everything by 2: a win will get you back your $2, plus $6 profit, for a total of $8.
What about 3–2 odds? Think this way: if you place a $2 bet and your horse wins, you get back your $2 plus $3 profit, for a total of $5. For a $1 bet, divide by 2: a win gets you your $1 back plus $1.50 profit (half of $3), totaling $2.50 (half of $5). Mathematically, bookmakers could just list those odds as 1˝–1, but I don’t think they ever do, because people are more comfortable with whole numbers.
Is the first number in betting odds always bigger? Usually it is, because most of the time any given horse is more likely to lose than to beat all the other horses. But not always: if a horse is thought to be more likely to win than to lose, the odds might be 1–2: a winning $2 bet would get you only $3, your original $2 plus $1 profit.
You may see these “backwards” odds described with the word “on”, so for example odds of 1–2 might be quoted as 2–1 on. Odds of B to P on and odds of P to B against are equivalent, whatever the numbers.
How do odds convert to probability? Well, theoretically the odds are P to B. So 3–1 odds would mean 3 chances out of 3+1 to lose (q = 3/4 = 75%) and 1 chance out of 3+1 to win (p = 1/4 = 25%). We can put this in a formula:
P–B odds or B–P odds on: pwin = B/(P+B), plose = P/(P+B)
I was careful to say “theoretically”, because bookmakers don’t take bets to break even: they want to turn a profit. So they try to quote odds somewhat less favorable to you than their guess about the horse’s actual chances.
What does this mean? When betting is involved, the probabilities will be different from what the formulas tell you. Specifically, your actual probability of winning — or the bookmaker’s estimate of your actual probability of winning — will be less than the formula says, and your actual probability of losing will be greater.
How does probability convert to odds? In the formula above, to go backwards just use your guess at pwin or plose. Since the two probabilities must sum to 1, you have two equations in the two unknowns, P and B:
pwin = B/(P+B)
1 − pwin = P/(P+B)
Unfortunately, those two equations are really the same equation rearranged, so there are infinitely many solutions for P and B pairs. That just reflects the fact that odds of 3–2 are equivalent to 1˝–1, 6–4, 9–6, 12–8, and so on.
But notice what’s the same about all of those equivalent P–B odds pairs: P is always 3/2 of B. Though we can’t use the probabilities to find specific odds, we can use the probabilities to find an P/B ratio — call it k — that corresponds to the 3/2 in this example.
Put P = kB, where k is the ratio we want to find. Use pwin = B/(P+B) and substitute kB for P:
pwin = B/(kB+B)
pwin = 1/(k+1)
k + 1 = 1/pwin
k = 1/pwin − 1
What does this mean? Theoretically, if you have an estimate for pwin, that corresponds to any odds P–B where P = 1/pwin − 1. If you think a horse has a 40% chance to win, then k = 1/0.40 − 1 = 2.5 − 1 = 1.5, so P should always be 1.5×W. That would be odds of 1.5 to 1, but to make the numbers nicer you multiply both by 2 and give the theoretical odds as 3–2.
Again, bookmakers are in business, so they will decrease the P/B ratio to give themselves a better chance at a profit. For instance, they might quote 5–4 instead of 3–2 (P/B ratio of 1.25 instead of the theoretical 1.5). You see the difference?
|Betting $4 on an event 40% likely to occur|
|Odds||P/B ratio||Your profit
if you win
|5 to 4||1.25||$5||+$4×0.6 − $5×0.4|
= $0.40 per bet
|3 to 2
(equals 6 to 4,
|1.50||$6||+$4×0.6 − $6×0.4|
= $0.00 (break even)
How much profit do bookmakers actually consider standard? I have no idea, but here’s an example. In True Odds Vs. Implied Odds, Odds Shark says that a bet on a National Football League event with a 50% chance of occurring will generally cost you $110 for a chance to win $100 (plus your $110 back). Effectively that’s odds of 10–11, which represents a P/B ratio of 0.909090… rather than the mathematically fair 1.0.
Odds Shark calls that a 10% profit for the bookmaker, but I don’t think they’re correct. Your bet will either win or lose, and in Odds Shark’s example those are equally likely. So for a $110 bet, the bookmaker has a 50% chance of keeping the $110 and a 50% chance of returning the $110 and paying the bettor’s profit of $100. The expected value to the bookmaker, per bet, is +$110×0.5−$100×0.5 = +$5, and $5/$110 is about 4.5%.