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I am reading this document. On page 22 they have given this figure enter image description here and then asked P(tea|88005 & organic)?

The formula they have used to find it is

P(tea|88005 & organic) = P(88005 | tea) P(organic | tea) P(tea) = .6(.8)(.5) = .24

But this question here says that we can also use the formula

P(tea|88005 & organic) = P(tea & 88005 & organic) / P(88005 & organic)

Which gives 0.2 / 0.2 = 1.

So then the two values that we have arrived at are different.

Which one should we use?

And which formula are they applying in the pdf? I mean which formula should we be using to calculate such conditional probabilities that involve more than 2 variables?

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Nikunj Banka
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1 Answers1

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This is just my intuition and I may very well be wrong, sorry in advance if that's the case.

I think you can immediately deduce that the person writing the pdf doesn't know what they are talking about, because on that same page they claim: P(tea|88005 & organic) = .24, and P(~tea|88005 & organic) = .05. Letting the event tea be A, and (88005 & organic) be B, this is the same as claiming it is possible that P(A|B) + P(~A|B) != 1, which by the rules of probability is clearly not possible.

I tried playing with the solution they gave, and found no way to arrive at it.

By conditional probability, your second solution should be the right one. That is, if you set A = tea, B = 88005, C = organic, P(A|B & C) = P(A & (B & C)) / P(B & C) = P(A & B & C) / P(B & C). This follows from the general formula for conditional probability.

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  • I agree that there are errors in the document. The document also seems to "round" probabilities which obscures the very important notion of an event's complement having a complementary probability. $$P(A) = 1 - P(\bar A) \ P(A|B) = 1 - P(\bar A|B)$$ – Axoren Feb 26 '15 at 20:11