Gujarati/ condidional probability

[fashepard]

Hi David,

I am having problems understanding some of the formulas in Gujarati chapter 2 on page 30.

2.7 page 30 P(A/B)= P(AB)/P(B) for me this always comes out to P(A)
When he explains it he refers to figure 2-1 ( c ) but then in the explanation I don’t think P(AB) is the numerator

Then in example 2.11 it gets worse for me
I understand the answer but not the formula

P(A/B) P(AB)/P(B) but P(A) P(B) =.6*.32 or .192
He gets .2 by dividing 100/500 which I understand but it is not P(AB) with a result of .625. I can get .625 if I use the formula on page 32 in example 2:12. I also am not sure how he gets form formula 2:10 to the one he uses in 2:12

Where am I going wrong?

Sorry for such a long question

Frank

 

[David Harper CFA FRM]

Hi Frank,

(Sorry for delay)

I *think* its the challenging notation of P(AB) which is the joint probability. It looks like you do see that:
joint probability P(AB) = P(A|B)P(B)
where P(A|B)P(B) = conditional probability * unconditional probability = joint probability.
This is the only different between Ex 2.11 and 2.12

Personally, I best cope with these using the trees. I dropped the Ex 2.11 into the tree below. (the accounting example).
http://learn.bionicturtle.com/images/forum/guj_ex211.png
You can see that the joint probability P(AB) = P(male and accounting) is given by the top node, far right. This 20% is equal to either/both:
100/500, or

P(A|M)P(M) = (100/300)*(300/500) = 100/500
where this should comport with the visuals of the tree

Let me know if that clears it up - I hope you can make Saturday, I will do a bit on "conditional" versus unconditional?

David

 

[fashepard]

I am stuck here P(A/B) P(AB)/P(B) but P(A) P(B) =.6*.32 or .192 is not P(AB) = P(A)*P(B)

 

[David Harper CFA FRM]

but P(A) P(B) =.6*.32 or .192 is not P(AB) = P(A)*P(B)

Right, but that only applies if A & B are independent. The fact that P(AB) does not equal P(A)*P(B) demonstrates they are not independent so only under independence can this be used P(AB) = P(A)*P(B). It can't be used here as male correlates with accounting.

 

[fashepard]

David

P(A|M)P(M) = (100/300)*(300/500) = 100/500 is the tree using the notation in Gujarati 100/300 is P(B/A). Not P(A/B) which is what you have above ,joint probability P(AB) = P(A|B)P(A). Also when you switch to A's and M's it is the opposite notion of Gujarati's notation. A=Male and B=accountant.. I guess I am not getting this is this Bayes Theorem? See you tomorrow in the web cast

 

[David Harper CFA FRM]

RIght, I had a typo in my original above, sorry, the joint just be be: P(AB) = P(A)P(B|A) = P(B)P(A|B),

as in either (using M and A above, only b/c that's less confusing to me)

joint P(MA) = p(M)*P(A|M) = p(A)*p(M|A); see how those are sort of intuitive...if M happens, them multiply by A conditional on M. And you can go either direction, so:

p(M)*P(A|M) = as you show, does not require Bayes and is more natural per the tree

but is equivalent to:
P(A)P(M|A)
where P(M|A) requires Bayes = (100/500)/(160/500)
so = (160/500)*(100/500)/(160/500) = 160/500*100/160= 20%...i.e., same result but uses Bayes.

David