Odds is confusing in a different way than some of the other terms in this series.
First, it’s a bit of an abstract concept, which I’ll explain below.
But beyond that, it’s confusing because it is used in everyday English as a synonym for probability, but it’s actually a distinct technical term.
I found this incorrect definition recently in a (non-statistics) book:
“…probability, which is the odds of something occurring… A fair coin has 50% odds of heads coming up on any flip.”
No. Just no.
It wasn’t the first time I’ve seen something similar.
And yes, I may have banged my head on the desk when I read that. (And I may have, in a spirit of kindness and being helpful, emailed the author to let him know how he could correct his definition. No response so far.)
We need to use the correct technical terms for probability and odds in statistics. So if you only understand one and think they’re the same, you’re going to be very confused in the realm of statistics.
Yes, they are both measuring the likelihood of an event. But they’re measuring it on different scales.
It’s just like degrees in Celsius and Fahrenheit.
They both measure temperature. But the scales are very different. You say it’s 30º out. I assume you meant Fahrenheit but you meant Celsius. I am going to be way, way overdressed.
Likewise, you tell me the probability (not the odds) of a fair coin coming up heads is .5.
The equivalent odds is 1.
If you tell me the probability is 1 when you mean odds (or vice versa), I will make inaccurate inferences about the coin.
That may not be a big deal, but if you’re a doctor telling me about how likely it is I will die in the next two months or develop a serious disease, there is a big, big difference.
If I read your publication about the likelihood that a student will drop out of school based on certain risk factors, I will come away with a very different conclusion if you don’t specify whether you’re reporting odds or probabilities and I assume wrong.
Let’s unpack this a bit. Say we do an experiment of 1000 coin flips in order estimate the likelihood of a head on any given flip. Let’s also assume this experiment came out perfectly and gave us exactly 500 heads and 500 tails.
Probability
The probability of a fair coin being heads is measured by the number of heads out of the number of flips:
So out of 1000 flips, we should get around 500 heads.
Pr(heads) = Number of heads/Number of flips
Pr(heads) = 500/1000
Most of us are pretty familiar with that definition and we’ve used it so long, that we can easily think in terms of probabilities.
Just like most of the world (outside the US) can think in terms of degrees Celsius.
Odds
Odds also measure the likelihood of the coin coming up heads.
But odds are expressed as the number of heads compared to the number of tails.
The odds of a fair coin coming up heads is measured by the number of heads compared to the number of tails (aka, non-heads):
So for every 500 heads, we expect 500 tails.
Odds of heads = Number of heads/Number of tails
It’s also often written as: Number of heads:Number of tails
Odds(heads) = 500/500 or 500:500
This can be simplified to: 1/1 or 1:1 or 1
So for every one head, we expect one tail.
Equal odds = 1
Equal probability = .5
The Takeaway
Be precise in the language you use in statistics.
Don’t use the terms likelihood, possibility, or probability when you mean odds. The first two are imprecise, and therefore, misleading. The third is incorrect.
If you’re reading something that is imprecise or looks like it might be incorrect, contact the author and ask for more information. You may or may not get a response.
Other articles in this series:
Series on Confusing Statistical Terms
Confusing Statistical Terms #1: Independent Variable
Confusing Statistical Terms #2: Alpha and Beta
Confusing Statistical Terms #3: Levels
Confusing Statistical Term #4: Hierarchical Regression vs. Hierarchical Model
Confusing Statistical Term #5: Covariate
Confusing Statistical Term #6: Factor
Confusing Statistical Term #7: GLM
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