What is the difference in purpose between expected values and Bayes' formula?

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Enlighten_me

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I know this isn’t of the LOS, but just wondering when do I know when to use expected values and Bayes’ formula when both are used for the purpose of revising/updating the probability when there is new information?
 
Hi,
Not sure I understand what you mean by using expected values to revise probabilities.
But Basically Bayes’s Formula states P( Event | Information) = P (Infomation | Event) x P (Event) / P (Information)
Let’s consider an example.
In a particular school, there are 54% boys and 46% girls. The girl students wear blue sweaters or red sweaters in equal numbers, while the boys all wear blue sweaters. An observer sees a (random) student from a distance, and all she can see is that the student is wearing a blue sweater. What is the probability that the student is a girl? (source: Wiley CFA Level 1Study Guide 2015, Volume 1, Page 201 => Btw I use Wiley and think it’s awesome!!)
step 1: What is the “Information” -> Student wearing a Blue Sweater (this has been confirmed)
step 2: We are given Pr (Boy) = 0.54, Pr(Girl) = 0.46, Pr(Blue | Boy) = 1 (as “boys all wear blue sweaters) , Pr(Blue | Girl) = 0.5 (as “girls wear blue/red in equal numbers”)
step 3: We want Pr(Girl | Blue)= ? = Pr(Blue|Girl) x Pr(Girl) / Pr(Blue)
We can work out Pr (Blue) = Pr(Blue|Boy) x Pr(Boy) + Pr (Blue| Girl) . Pr (Girl). So from here it’s just plug and play into Baye’s formula.
However, I prefer simplifying this problem into a tree diagram.
Basically first branches- > Girl or Boy with probabillities 0.46 and 0.54
second branches (two for each of the two first branches )-> Blue or Red with probabilities 0.5 and 0.5 for where first branch is Girls, or probabilities1 and 0 where first branch is Boys
(note the second branches show conditional probabilities while the first branches are unconditional probabilities)
now to answer “probability of Girl given Blue” work backwards from the diagram..All you have to do is focus on the branches that end with Blue….multiply along each of these branches and record the probabilities…so branch (Girl, Blue) -> 0.46 x 0.5 = 0.23, branch (Boy, Blue) -> 0.54 x 1 = 0.54, Total Blue = 0.23 +0.54 = 0.77
So given Blue, the probability of that being a girl is simply 0.23/0.77 = 0.2987
Once you’ve worked out conditional probability, any conditional expectations are simply computed by substituting conditional probabilities in the usual Expected Value formula = possible value of random variable x probability of that value
 
Iampossible wrote:
Hi,
Not sure I understand what you mean by using expected values to revise probabilities.
But Basically Bayes’s Formula states P( Event | Information) = P (Infomation | Event) x P (Event) / P (Information)
Let’s consider an example.
In a particular school, there are 54% boys and 46% girls. The girl students wear blue sweaters or red sweaters in equal numbers, while the boys all wear blue sweaters. An observer sees a (random) student from a distance, and all she can see is that the student is wearing a blue sweater. What is the probability that the student is a girl? (source: Wiley CFA Level 1Study Guide 2015, Volume 1, Page 201 => Btw I use Wiley and think it’s awesome!!)
step 1: What is the “Information” -> Student wearing a Blue Sweater (this has been confirmed)
step 2: We are given Pr (Boy) = 0.54, Pr(Girl) = 0.46, Pr(Blue | Boy) = 1 (as “boys all wear blue sweaters) , Pr(Blue | Girl) = 0.5 (as “girls wear blue/red in equal numbers”)
step 3: We want Pr(Girl | Blue)= ? = Pr(Blue|Girl) x Pr(Girl) / Pr(Blue)
We can work out Pr (Blue) = Pr(Blue|Boy) x Pr(Boy) + Pr (Blue| Girl) . Pr (Girl). So from here it’s just plug and play into Baye’s formula.
However, I prefer simplifying this problem into a tree diagram.
Basically first branches- > Girl or Boy with probabillities 0.46 and 0.54
second branches (two for each of the two first branches )-> Blue or Red with probabilities 0.5 and 0.5 for where first branch is Girls, or probabilities1 and 0 where first branch is Boys
(note the second branches show conditional probabilities while the first branches are unconditional probabilities)
now to answer “probability of Girl given Blue” work backwards from the diagram..All you have to do is focus on the branches that end with Blue….multiply along each of these branches and record the probabilities…so branch (Girl, Blue) -> 0.46 x 0.5 = 0.23, branch (Boy, Blue) -> 0.54 x 1 = 0.54, Total Blue = 0.23 +0.54 = 0.77
So given Blue, the probability of that being a girl is simply 0.23/0.77 = 0.2987
Once you’ve worked out conditional probability, any conditional expectations are simply computed by substituting conditional probabilities in the usual Expected Value formula = possible value of random variable x probability of that value
Click to expand...
Hi I really appreciate your lengthy example on the Bayes’ formula… i can’t seem to remember the correct formula which got me to review the topic. Thank again
 
No worries mate, happy to help where I can.
Suppose no one told you about Bayes’ Theorem and gave you this problem at high school you would most likely tackle it by drawing a simple tree diagram.
So at the exam, even if you forgot Bayes’ formula, you can solve most questions of this nature using a tree diagram.
The only question which you cannot answer might be the type that says “Bayes formula states that…..” and asks you to recall the exact formula (or a rearranged form of it).
If you like to link stuff, you may already have noticed that: P(AB) = P(A|B). P(B) = P(B|A).P(A) (Multiplication Rule)
=> P(A|B) = P(B|A) x P(A)/P(B) (Bayes’ formula)
Replace A with “Event” and B with “Information” and you have it in the form applied to questions. No need to memorize.
Cheers and all the best !!
 
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