Total Probability Rule definition, understand, makes, total, would

MeteorShower

Does anyone have an easy way of explaining the total prob rule?

I understand how to get the answer and the tree diagram in my notes makes sense but the wording of the definition throws me off. If anyone has a easy way of explaining it I would appreciate it.

Thanks!

spartanag07

Ok I will try....

Total probability is basically the probability something will happen, considering all the different pathways through which that something can occur (and weighting them accordingly given their probability.)  

Let's say I am a soccer player and want to work out the chances of me scoring a goal. I have 2 pairs of boots - red ones and blue ones. I can't wear both pairs at once (the scenarios are mutually exclusive) and I will always play wearing one of the pairs of boots (the scenarios are exhaustive).

Therefore, the total probability of me scoring a goal = (the probability of me scoring a goal in red boots x the probability of me wearing the red ones) + (the probability of me scoring wearing blue boots x the probability of me wearing the blue boots).

If I had 3 pairs of boots, the same logic would apply.  

Hope that helps

JPSem

That's an excellent explanation man... you know your stuff very well kiakaha

ramdabom

Thanks jonnykay...if someone could have a crack at explaining Bayes formula that would be awesome, it's still tripping me up!
Cheers

marsilni

not sure if this helps... (I am writing this from memory) .

instead of memorizing the formula - draw out the tree and calculate the total probs. and then work backward.s.. the Bayes formula is the only thing that is intimidating.

company has a 80% chance of product launch success, 20% chance of product launch failure.
if product launch is successful - company makes profits 80% of the time. If failure - makes profits 30% of the time.

Company has made a profit. What was the probability that the product launch failed.


draw a tree diagram

Product launch success 0.8 profit 0.8
success 0.8 loss 0.2

launch failure 0.2 profit 0.3
0.2 loss 0.7

company made a profit...

success/profit=0.8 * .8 = .64
success/loss = 0.16
failure/profit = 0.06
failure/loss = 0.14

Total profit probability = .64 + .06 = 0.7
This is a given... probability that it was a failure on the product launch = 0.06 / 0.70 ...

CP

Ionutzakis

Thanks very much cpk123... I agree there are other ways to solve these problems, it's just bugging me that I can't seem to get the formula, seeing as I'm pretty comfortable with the rest of the probability stuff.
But you're right, I might forget about it and solve using the method you've described -- one less formula to remember isn't exactly a bad thing

manasib

So helpful! Thank you so much.. It's not to bad when you put it into every day language.

Bluetick1010

> success/profit=0.8 * .8 = .64
> success/loss = 0.16
> failure/profit = 0.06
> failure/loss = 0.14
>
WRITING THIS IN CAPS TO HIGHLIGHT STUFF
NOTE THIS POINT HERE --- FORGOT TO WRITE THIS BEFORE
SUM OF ALL THE PROBABILITIES OBTAINED HERE SHOULD BE = 1.

YOU CANNOT GET MORE THAN OR LESS THAN 1. IF YOU DO - YOU ARE MAKING A MISTAKE SOMEWHERE. REMEMBER THESE ARE MUTUALLY EXCLUSIVE EVENTS - EITHER A SUCCESS OR A FAILURE AND UNDER EACH CASE - EITHER PROFIT OR LOSS.

YOU MIGHT HAVE a BIGGER TREE --- 3 options maybe... but even then sum of probabilities at the FINAL node level should be 1.

Additionally - P(SUCCESS) = 0.8

P(success/profit) =0.8 * .8 = .64
P(success/loss) = 0.16

SUM of the ABOVE two = 0.8 = P(SUCCESS)

Again a check to see if you are working in the right direction.

Cheers

CP

ftwcfa

Kiakaha Wrote:
-------------------------------------------------------
> Thanks jonnykay...if someone could have a crack at
> explaining Bayes formula that would be awesome,
> it's still tripping me up!
> Cheers

bayes formula gives the probability of event A happening given that B happened USING THE FACT that we know the probability of B happening given A happened.

bayes formula is as follows:

P(A|B)*P(B) = P(B|A)*P(A)

note that the above equation = P(A and B) because P(A|B) = P(A and B)/ P(B) and P(B|A) = P(A and B)/ P(A).

I will illustrate with a simple example.

suppose the probability of someone passing CFA level I given they studied 300+ hrs is 0.5. The probability of someone studying 300 +hrs is 0.25. we also know the probability of someone passing Level I is about 0.38.

lets look at what we know:

P(Pass | 300+hrs of study) = 0.5, P(300+hrs of study) = 0.25 and P(Pass) = 0.38.

now we ask the question, what is the probability someone studied 300+hrs GIVEN that they passed the exam, P(300+ | Pass) = ?

we use the formula to solve

P(Pass | 300+ hrs)*P(300+hrs) = P(300+ | Pass)*P(Pass)
0.5*0.25 = P(300+|Pass)*0.38.

P(300+|Pass) = 0.33.

note carefully the difference between P(Pass | 300+) and P(300+|Pass). they are NOT always the same.



Edited 1 time(s). Last edit at Thursday, February 17, 2011 at 07:34PM by BayStreet.

agulani

Thanks very much baystreet! Slowly starting to get it