Colloquial language doesn’t make this distinction, but by technical convention, they are different.
Specifically, ‘odds’ refers to expressions like ‘5 to 3 against’; numerically, that’s the fraction 5⁄3, or rather (because of the ‘against’) its reciprocal, 3⁄5. Thus odds run from 0 (impossible) to infinity (certain), with odds of 1 being perfectly balanced between Yes and No. In contrast, probabilities run only from 0 to 1. An event with odds of 5 to 3 against, or equivalently odds of 3⁄5, has a probability of 3/(3+5) = 3⁄8. So the numbers are different. The conversion formulas are O = P/(1 − P) and P = O/(1 + O).
Then there are log-odds; this is log₂ O bits. (You can also use other bases than 2 and correspondingly other units than bits.) Now 0 indicates perfect balance between Yes and No; a positive number means more likely Yes than No, and a negative number means less likely Yes than No. Log-odds run from negative infinity (impossible) to infinity (certain).
Specifically, ‘odds’ refers to expressions like ‘5 to 3 against’
Oh right, I forgot about that definition. The main probability conversions that I was aware of involved converting between fractions and percentages, sometimes expressed instead as probabilities between 0 and 1. Theoretically, it makes sense that odds can also be converted to or from probabilities, now that I think about it. Thanks for your explanation.
Odds can be expressed as a ratio of two numbers [or] as a number, by
dividing the terms in the ratio [....] Odds range from 0 to infinity, while
probabilities range from 0 to 1 [...]”
Yes, that’s exactly what I said. There is no way to express a fraction greater than 100% using odds notation; Saying that odds are “1 million to 1” is 99.9999%, still under 1.
In the Wikipedia
article, take a
look at the table below the words “These are worked out for some simple odds”.
The odds that
TobyBartels is talking
about,
which one gets by dividing the
numbers in an “n to m” expression, and which go from zero to infinity, are
shown in the second and third columns of that table (o_f and
o_a). Probabilities, which go from 0 to 1 or 0% to 100%, are shown in the
fourth and fifth columns (p and q).
Did you actually read the article you linked? It says the exact same thing as I did, phrased differently. Their “Odds range from 0 to infinity” means that any number from 0 to infinity can be used in the odds ratio, but still always represent a probability between 0 and 1. Which is precisely what I said.
Um, representing a number between 0 and 1 is not the same as being a number between 0 and 1. The representation of p = 3⁄8 as odds = 3⁄5 (“5 to 3 against”) is useful in practice, for example because bayes’ rule reduces to plain multiplication for odds ratios.
Colloquial language doesn’t make this distinction, but by technical convention, they are different.
Specifically, ‘odds’ refers to expressions like ‘5 to 3 against’; numerically, that’s the fraction 5⁄3, or rather (because of the ‘against’) its reciprocal, 3⁄5. Thus odds run from 0 (impossible) to infinity (certain), with odds of 1 being perfectly balanced between Yes and No. In contrast, probabilities run only from 0 to 1. An event with odds of 5 to 3 against, or equivalently odds of 3⁄5, has a probability of 3/(3+5) = 3⁄8. So the numbers are different. The conversion formulas are O = P/(1 − P) and P = O/(1 + O).
Then there are log-odds; this is log₂ O bits. (You can also use other bases than 2 and correspondingly other units than bits.) Now 0 indicates perfect balance between Yes and No; a positive number means more likely Yes than No, and a negative number means less likely Yes than No. Log-odds run from negative infinity (impossible) to infinity (certain).
Oh right, I forgot about that definition. The main probability conversions that I was aware of involved converting between fractions and percentages, sometimes expressed instead as probabilities between 0 and 1. Theoretically, it makes sense that odds can also be converted to or from probabilities, now that I think about it. Thanks for your explanation.
‘5 to 3 against’ is 3⁄8, not 3⁄5. Odds of ‘N to M’ or ‘N to M against’ are always between 0 and 1.
5 to 3 against is 3⁄5 (as odds), which is a probability of 3⁄8. You are muddling probability and odds ratios in an unacceptable way.
Wikipedia:
Yes, that’s exactly what I said. There is no way to express a fraction greater than 100% using odds notation; Saying that odds are “1 million to 1” is 99.9999%, still under 1.
In the Wikipedia article, take a look at the table below the words “These are worked out for some simple odds”. The odds that TobyBartels is talking about, which one gets by dividing the numbers in an “n to m” expression, and which go from zero to infinity, are shown in the second and third columns of that table (o_f and o_a). Probabilities, which go from 0 to 1 or 0% to 100%, are shown in the fourth and fifth columns (p and q).
You said ‘Odds […] are always between 0 and 1’, while Wikipedia said ‘Odds range from 0 to infinity’, so you didn’t say the same thing.
Did you actually read the article you linked? It says the exact same thing as I did, phrased differently. Their “Odds range from 0 to infinity” means that any number from 0 to infinity can be used in the odds ratio, but still always represent a probability between 0 and 1. Which is precisely what I said.
No, that’s not what you said. I am now done with this conversation.
Um, representing a number between 0 and 1 is not the same as being a number between 0 and 1. The representation of p = 3⁄8 as odds = 3⁄5 (“5 to 3 against”) is useful in practice, for example because bayes’ rule reduces to plain multiplication for odds ratios.