Can you get into the mindset of a probability?

You thought you had a simple and easy life, roaring along the highway from 0 upwards.

Until you see this ahead of you.
You have a twin sister, who in her youth was similar to you in many respects.
But she was always more vivacious and sporty.

When you read comic books, she preferred to dance.

When you watched TV, she joined a cycling club.

Now you are almost at a dead end.
And she is running free.
But you keep in touch.

There is a 1 to 1 correspondence.

Every point in your life matches exactly one point in her life. It's just that your steps are getting smaller and smaller, and hers seem to be getting bigger and bigger.

Does that give you a feel for Prob and Odds?
(In my terminology)

Probability = risk = chance

To me, these are simply synonyms. If people find example where they are different (within medical stats or math) please let me know.
Odds is (are?) different from probability.

Odds is Prob / (1-Prob)

or you could state it as

Prob of happening / (Prob of not happening)

or

Prob (win) / Prob (lose)
The two are FREELY INTERCONVERTIBLE.

You can always, without exception, deduce either from the other.
The questioner is asking a more subtle question which pertains to odds RATIOS.

They are NOT freely interconvertible with risk ratios (probability ratios, also called relative risk).
Why are Odds and Probability (risk) freely interconvertible and yet Odds RATIOS and Risk RATIOS not?

Insight provided by Dr Pabari in the Cath Lab of the Hammersmith.

(She is an imager doing TOE for TAVI which is why she is so clever)

Odds are related to probability in a curved way

This is why knowing the ratio of odds does NOT give you the ratio of probabilities

Nor vice versa.

If the above is not clear watch this
pscp.tv/w/baB8yDFxTFFH…
Why do we bother to even think about odds if they are such a pain?

Pacemaker infections.

pscp.tv/w/baB9QzFxTFFH…
Odds Revision Questions C
------------------------------

Third time lucky!

Let's aim for 100% this time, so that we can move on to more harder-er stuff this weekend.

Here is the Francis Industries patented
"Probability <--> Odds convertometer"
Probability of 0.5, means odds of what?

Hint. The "A" is there for a reason
And an odds of 9 is a probability of what?
Hint:
Now look at these two points C and D.
What is the relative risk of D, by comparison to C?

In other words, what is the probability of D divided by the probability of C?
Because at that region, the curve is almost a straight line, the ratio between the ODDS of D and the ODDS of C, i.e. "the Odds Ratio", is also almost exactly the same as the above ratio you have calculated.
Now look at these points, E and F.
What is the RISK RATIO (also called relative risk) of F, by comparison to E?
And, finally, what is the ODDS ratio of F by comparison to E?
So D by reference to C has RISK RATIO 2
And F by reference to E has RISK RATIO 2

Whle D by reference to C has ODDS RATIO ~2
And F by reference to E has ODDS RATIO much greater than 2!
So if I tell you "Patients scoring 'high-risk' on the Francisometer have a Relative Risk of 2.0 for all-cause mortality, compared with those scoring 'low-risk' on that test.",

What is their ODDS RATIO?
... because we don't know where they are on the curve.

If they are at the left, all rather low risk, then the answer is "just over 2".
But if they are at the right, then the answer is "much more than 2".
For nerds who want the exact Odds Ratios for the above:

Odds Ratio for D versus C is:
(0.2/0.8)/(0.1/0.9)=2*9/8=9/4 = 2.25 (so "Just over 2" is correct)

Odds Ratio for F versus E is:
(0.8/0.2)/(0.4/0.6) = (4)/(2/3) = 12/2 = 6 (so "About 5" is correct)
Question from @veitchemma
I will discuss tomorrow the reasons why Odds Ratios are statistically/mathematically preferable over Risk Ratios (relative risks).

However, people in general can understand Risk Ratios but not Odds Ratios.

This Twitter thread has proven this to me more starkly than I imagined!
All I can offer is this.

If the probabilities are all low, e.g. less than ~10% or so, the odds ratio is only a little higher than the risk ratio. So the average person can simply treat them as almost synonymous.
But as the probabilities rise (i.e. moving to the right on the Odds-v-Prob curve), the Odds Ratio becomes much higher than the Risk Ratio.

There is no easy fix for that.
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