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!
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
That’s an excellent explanation man… you know your stuff very well kiakaha
Thanks jonnykay…if someone could have a crack at explaining Bayes formula that would be awesome, it’s still tripping me up! Cheers
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 …
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
So helpful! Thank you so much… It’s not to bad when you put it into every day language.
> 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
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.
Thanks very much baystreet! Slowly starting to get it
That was great BayStreet! @Kiakaha: Let me write down what I understood too… (it will probably help me too) The most important thing to remember: Conditional Probability = P(A|B) = P(A and B)/P(B) Since P(A and B) = P(B and A) (Probability that both A and B happen), P(A|B)*P(B) = P(B|A)*P(A) Now let’s say, when I am given an Econ question, I take either less than a min to answer it or 1 to 2 mins to answer it or more than 2 mins to answer it. P(less than a min) = .45 P(1 to 2 min) = .30 P(more than 2 min) = .25 Ofcourse all these case will add up to 1 since they mutually exclusive and exhaustive. (.45+.3+.25 = 1) Now probability of my answer being correct is related to how long I took to answer the question. Assuming that the more time I take, the more unsure I am of the answer, P(Correct | less than a min) = .75 P(Correct | 1 to 2 min) = .20 P(Correct | more than 2 min) = .05 What was the probability of Correct answer? Using the Total Probability Rule you explained beautifully, .75*.45 + .20*.30 + .05*.25 = .41 or 41% Now starts the use of Bayes’ Formula: I was given a question and I find that my answer is correct. Answer the following questions: 1. Can you tell me what is the probability that I took less than a min to answer it? i.e. I need to know P(less than a min | Correct) Using Bayes’ Formula, P(less than a min | Correct) * P(Correct) = P(Correct | less than a min) * P(less than a min) So, P(less than a min | Correct) * .41 = .75 * .45 = .82 So it is 82% probable that I took less than a minute to answer it. 2. Can you tell me what is the probability that I took 1 to 2 mins to answer it? Using Bayes’ Formula, P(1 to 2 min | Correct) * P(Correct) = P(Correct | 1 to 2 min) * P(1 to 2 min) So, P(1 to 2 min | Correct) * .41 = .20 * .30 = .146 So it is 14.6% probable that I took 1-2 mins to answer it. and so on… If you understand this completely, you have got the Ex13 on page 469 of the curriculum. My suggestion for what its worth: Don’t get lost in old/new info/event etc… Just approach them as A, B and use the conditional probability formula given above.
Anish, this finally makes sense to me!! thanks so much!