1:3 odds versus 33% chance

Factual Questions
1

I see both Biden and Saunders are rated at a “1 in 6” chance of winning the nomination outright. (On the 538 site.)

How is this different from “fifteen point something percent chance?” Why are the numbers stated this way? Perhaps it has to do with the Nate Silver’s background in sports betting. But then why do sports bettors use one way and not the other.
Why not express it as a “1/6 chance,” using fractions?
In writing we use fractions for accuracy, and percentages for ease of reading. I cannot recall the last time I used the 1-in-6 form in writing.

2

I don’t know. It sounds perfectly colloquial to me. That’s the way I express probability. You can also use the form “five to one odds,” but that can be confusing to people who are not used to that construction (five to one odds is the same as one in six probability.)

3

Oh, and 1:3 odds would not be a 33% chance. 1:3 odds conventionally would mean 75% probability of an outcome happening. 3:1 would be 25% probability. (Typically odds are stated to mean “odds against.”)

4

Can I express “3 to 1” as “a $3 bet will pay $4?”
What is really rough is a construction like “6 to 5.” I find this confusing.

5

Every dollar pays 3, so it would pay $9.

6

For some reason, the odds of a binary test (i.e. succeed/fail) are quite often expressed as “successes in attempts” when talking about gambling or contests between opponents. It’s more intuitive for people who aren’t mathematically inclined or who may have an irrational dislike for fractions (often dating back to their school days).

1 in 6 is easily understood that in six attempts, they would succeed once. 1/6 chance may be more concise or mathematically accurate, but those traits are less useful when trying to clearly convey information to as wide a range of readers/viewers as possible.

7

Stating odds does get confusing! But “1 in 6”, “⅟6”, and “16.7/100” are all synonyms; and “16.7%” is a centuries-old abbreviation for the latter. (Pedants who point out that 1/6 ≠ 16.7/100 are dismissed from class.)
A bookie offering true odds on a 1-in-6 proposition might quote the payoff as “5 to 1” or “6 for 1” but bookies generally do NOT offer true odds, so you might be quoted “9 to 2” or even just “4 to 1” for the payoff. (Nate Silver is trying to quote true odds, rather than booking bets, so doesn’t use the “to” or “for” constructions.)

(What gets VERY confusing are “American decimal odds,” or whatever they’re called. There’s a fair chance the explanation following is wrong!
A 5-to-1 payoff would be quoted as “+500”; a 1.2-to-1 payoff would be quoted as “+120.” So far, so good. But what if you’re betting an odds-on favorite; e.g. laying the 1.2-to-1 bet. This is a 0.833-to-1 payoff so should be quoted as +083, but the leading zero is disallowed in “American decimal odds”, so this would be quoted as “-120”. :smack:)

8

I think an issue is that election results are commonly expressed in percentages, and a lot of people, consciously or subconsciously, will assume that the percentage they’re looking at is a prediction for a percentage of the vote. Their 2016 forecast for the general election ended with a 70% chance of a Clinton victory, and a lot of people’s reactions after Trump’s victory sure sounded like they thought a 70% chance meant a prediction of a landslide, rather than a prediction that it will be close.

9

Lord Feldon has it basically right. The idea is that the average person doesn’t process percentages the same way as they do simple odds. They spent a lot of time near the end of the 2016 election converting the percentages to ratios so people would understand them better. So, when they next predicted an election (in 2018, I believe) they started displaying the ratios first, with the percentages smaller.

It’s not so much that people mixed it up with the percentage of the vote, but just that people have a tendency to misunderstand percentages. People treat stuff like an 85% chance as certain, rather than a 5 in 6 chance (or a 1 in 6 chance of failure).

10

This depends what country you are in. Some countries take 3:1 to mean I give you $1, if I win, you give me $3 ($3 pays $9).

Other countries take 3:1 to mean I put up $1, you put up $3, whoever wins takes $4. ($3 pays $12)

11

Just by the way, Biden is now at 14:15, or 93.333%.
Seriously, I find this confusing. I bet $14 to win $15?

12

The opposite. Bet $15 to win $14.

It happens sometimes in horse racing when a horse is an overwhelming favorite (or at least thought to be so) that you win less than you bet.

13

Thinking on this again, those numbers don’t add up, wherever you got them.

93.3333% is very, very likely, so the odds should be lopsided. Odds of 14:15 are nearly even money or pretty close to 50% instead.

Maybe you were looking at a probability and interpreted as odds (or vice-versa)?

If it was a probability:

14 / 15 is 93.33333%

A probability of 14/15 as odds is 1:14 or betting $14 to win $1.

If it was actually odds:

14:15 odds is 15/29 probability = 51.72%

You are betting $15 to win $14.

ETA: Ah, I see those numbers came from fivethirtyeight. They are reporting probabilities NOT odds. 14 in 15 is 14/15 = 93.3%. As odds, that’s 1:14. They are reporting probabilities. Not sure why they use that form but it may be to give people an idea of likelihood. People are bad with straight numbers (example - my parents consider an 80% rain forecast as a foregone conclusion but would not consider a 4 in 5 chance to be the same).

14

Thinking even more on it, maybe a primer on odds vs probability is in order, considering some of the posts in this thread.

Odds and probability measure the same thing in slightly different ways.

Odds tells you “ways to succeed” vs “ways to fail”

Probability tells you “ways to succeed” vs “total number of ways”

If we have a coin, the ODDs of flipping heads is 1:1, i.e. there is 1 way to flip heads and 1 way NOT to flip heads.

The PROBABILITY of flipping heads is 1/2, i.e. there is 1 way to flip heads and 2 total ways the coin can flip.

So, 1:1 odds is the same as 1 in 2 probability.

If we have a single 6-sided die, the ODDs of rolling a ‘6’ is 1:5, i.e. there is 1 way to roll the 6 and 5 ways NOT to roll the 6.

The PROBABILITY of rolling a 6 is 1/6, i.e. there is 1 way to roll the 6 and 6 total possible rolls.

So, 1:5 odds is the same as 1 in 6 probability.

You can see how these are expressing the same uncertainty but in different forms. You can also see how the odds construction would be relevant for betting. 1:1 odds (or 50% probability) tells you it’s an even money bet, i.e. betting $1 or losing $1 are equally likely. 1:5 odds tells you to get even money, you would need to win 5 times as much for rolling a 6 as the amount you lose for not rolling it.

Note all of this is not directly relevant to the 538 website. They are reporting probabilities (NOT odds). But instead of using a percentage, they are using a different presentation method.

We can only guess why but it is probably to give people a better sense of scale. Tell somebody 0.1%, and they’ll know it is low but not how low. Tell them 1 in 1000 and that usually does the trick.

15

If you’re looking at 538, Biden is not listed as 14:15. He has a 14 in 15 chance of winning. Those are different. In US style odds the probability 14 in 15 would be expressed as 1:14, not 14:15. For every $14 staked, you can win your original bet back plus one dollar. This is also expressed as “14 to 1 on”.

That said, yes, it does kinda work out that way. If you bet $14 at a bookie offering 1:14 and Biden wins the nomination, you will get your $14 back plus a buck for winning, making it $15. But you’ve only really won $1, not $15.

ETA: I see Great Antibob has more than ninja’d me. That’s what I get for leaving this page open for so long without refreshing.

16

Don’t you mean 5:1 here? 1:5 would mean 5 in 6 probability. At least I’m used to typically hearing odds expressed as “odds against” unless the word “on” follows. So “five to one odds” would mean bet a buck to win five, or a 1 in 6 probability (not accounting for the bookie’s take, where odds wouldn’t exactly follow probability.)

17

I am getting these numbers from 538. They are saying Biden is a 14 in 15 (94%) favorite. So if I bet a dollar and old Joe wins, what do I get?
As I said, I am feeling really dumb about this.

18

You would get your dollar back and a profit of seven cents.

19

And, yes, this isn’t necessarily all that intuitive. 14 in 15 translates into 1:14 odds. That means, you bet $14 to win $1. So, if you bet just $1, you would win 1/14 of a dollar, or seven cents. (Well, like 7.14285714285 cents, repeating, to be exact.)

(Of course, since the house needs to make some money, they will give you a little bit worse odds if they feel the probability is truly 14 in 15, but I’m answering this question assuming odds given reflect probability.)

20

Thank you all.