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I understood the answer to my problem here when I substituted the "given that" symbol with the phrase "out of"

I got this idea from 3Blue1Brown where Grant points out that people are less confused when this phrase is used.

So why do we persist with using "given that"?

As a software developer I am used to using "given that" in pseudo code for unit tests, so the phrase especially threw me.

[Update]

I realise now that I am falling into the base-rate fallacy and am looking for an easy way to remember how to avoid it. I have often heard the "conditioning on" symbol read as "given that" and when that happens I get confused. For example in "“Given that it’s cloudy, the probability of rain is high” when we write it as P(Rain|Cloudy) I tend to read this "P(Rain and Cloudy) given it is cloudy" So I mistakenly think that we already know that it is cloudy (i.e we are given this) so we just need the probability of rain.

[Update]

I reverted to my original question with clarification in brackets. To hopefully make the question read better.

[Update]

Wikipedia mentions that

"the conditional probability of A given B" ... can also be understood as the fraction of probability B that intersects with A

Kirsten
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    What do you understand "out of" to mean?? In the CV post you reference, no answer has used the phrase "given that." – whuber Oct 19 '21 at 21:56
  • I understand it to mean "divided by". I used it in an update to the end of my question. https://stats.stackexchange.com/questions/548583/stuck-on-a-step-calculating-naive-bayes-classifier – Kirsten Oct 20 '21 at 00:03
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    "Given that" is to be understood as "conditional on" – Xi'an Oct 20 '21 at 13:11
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    It sounds like you might be misreading the notation for conditional probability as if it represented division. See https://en.wikipedia.org/wiki/Conditional_probability for some clarification. – whuber Oct 20 '21 at 15:27
  • I am reading it as the probability of both divide by the probability of the one we condition on because I keep falling into the base rate fallacy. I am looking for an easy way out. https://en.wikipedia.org/wiki/Base_rate_fallacy – Kirsten Oct 20 '21 at 20:34
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    I would like to see this question spelled out more fully instead of references to other questions. What is the full expression of Bayes theorem with the phrase 'given that' that the phrase 'out of' is supposed to replace? – Sextus Empiricus Oct 23 '21 at 23:50
  • For instance you give the example *“Given that it’s cloudy, the probability of rain is high”*. But I do not get two things here. 1) What is the connection with Bayes Theorem 2) When I replace the phrase I do get something that makes no sense: “Out of it’s cloudy, the probability of rain is high” – Sextus Empiricus Oct 23 '21 at 23:52
  • @SextusEmpiricus Thank you. I am struggling with how to ask things. I had another go at ~https://stats.stackexchange.com/questions/549502/is-there-a-tabular-or-graphical-way-to-show-the-updating-of-a-belief – Kirsten Oct 24 '21 at 02:00

2 Answers2

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Suppose the probability of the Kansas City Chiefs making the playoffs this season is about $72\%$ and the probability of them winning Super Bowl LVI at the end of the season is about $5\%$, then (since they must make the playoffs to progress to the Super Bowl) you can say

"the probability of the Kansas City Chiefs winning Super Bowl LVI given that they make the playoffs is about $7\%$"

but since this season only happens once you cannot say "out of" in a meaningful sense

Henry
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  • I don't see why not. The season only happens once, but you can consider the multiple possible futures where the Chiefs make the playoffs. – Accidental Statistician Oct 23 '21 at 21:42
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    @AccidentalStatistician - perhaps in your model or in your multiverse thought experiment. But the 2021-22 season is not going to repeat itself in reality. Using "out of" also seems to require a set of equally likely potential events, and probability often does not work that way. – Henry Oct 24 '21 at 00:14
  • I don't the former is a problem, but maybe because I lean more towards the Bayesian conception of probability. The part about "out of" implying equally-likely events makes sense, though. – Accidental Statistician Oct 24 '21 at 00:34
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As a software developer I am used to using "given that" in pseudo code for unit tests, so the phrase especially threw me.

First of all, different disciplines use different terminologies. That includes sometimes using same words differently.

But taking this aside, in BDD the tests are written in given-when-then, where

The given part describes the state of the world before you begin the behavior you're specifying in this scenario. You can think of it as the pre-conditions to the test.

Conditional probability is also about focusing (conditioning) on a specific scenario. “Given that it’s cloudy, the probability of rain is high” or “given that the ground is wet, there’s elevated probability that it rained.” Same as the test functions in a context, here we look at the probability of an event in a context.

Tim
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