 # Bayes' theorem

Could someone please explain how they understand and remember Bayes’ theorem? I see examples and can follow the explanations but I don’t really understand the theorem and therefore can’t remember it. I would really appreciate it.

jeez…glad u brought this up. I totally forgot about it myself

I would like to go over this too

it would be easier if you know how to tackle the problem using a tree diagram.

The way i remember it is that it is derived from p(A and B) = p(A|B) * p(B) = p(B|A) * p(A) So Bayes theorem is (p(A|B) = p(b|A) * p(A) ) / p(B)

yeh, i NEVER remember the formula… i think the best way to remember it and work it out is to draw a tree diagram… plus, it becomes very intuitive

I agree w/ sv102307, that’s usually how i think of it when needing to derive it. prob of both A and B = p(A and B). If p(A|B) is probability of finding A given B, then both are true only if B is true AND A-given-B is true. Therefore p(A and B) = p(A|B)*p(B). You can do the same thing by switching A and B to get p(A and B)=p(B|A)*p(A). Equating these gives p(A|B)p*p(B) = p(B|A)*p(A) Or, finally, p(A|B) = p(B|A)*p(A)/p(B)

has anyone got a question we could try and work out? if you see the tree-diagram process of working it out, you’ll see that its actually very easy

Suppose a certain drug test is 99% accurate, that is, the test will correctly identify a drug user as testing positive 99% of the time, and will correctly identify a non-user as testing negative 99% of the time. This would seem to be a relatively accurate test, but Bayes’ theorem will reveal a potential flaw. Let’s assume a corporation decides to test its employees for opium use, and 0.5% of the employees use the drug. We want to know the probability that, given a positive drug test, an employee is actually a drug user.

0.3322 ?

To directly use the formula, i first just write out the probabilities as below: p(positive|user) = .99 p(not positive| not user) = .99 we need to calculate p(user|positive) = (p(positive|user) * p(user)) / p(positive) p(positive) = p(positive| user) * p(user) + p(positive|not user) *p(not user) = .99 * .005 + .01 * * .995 = .0149 p(user|positive) = (.99 * .005) / .0149 = .3322

barthezz Wrote: ------------------------------------------------------- > Suppose a certain drug test is 99% accurate, that > is, the test will correctly identify a drug user > as testing positive 99% of the time, and will > correctly identify a non-user as testing negative > 99% of the time. This would seem to be a > relatively accurate test, but Bayes’ theorem will > reveal a potential flaw. Let’s assume a > corporation decides to test its employees for > opium use, and 0.5% of the employees use the drug. > > > > We want to know the probability that, given a > positive drug test, an employee is actually a drug > user. answer is .3322 and a question from wikipedia for further reference.

using tree diagram Drug User (.005) ------ positive (.99) ------ negative (.01) Non Drug User (.995) ------ positive (.01) ------ negative (.99) So P(Drug user|positive) = (.99 X .005) / {(.99 X .005)(.01 X .995)}

Bayes theorem may be in the curriculum, but I would be more worried about it’s theoretical component than actually doing a computation. Probability is over-rated at level 1. Focus on hypothesis testing and the standard calculations (standard deviation, variance, standard normal distribution, simple regression). Those are the money shots. If you don’t have Bayes theorem down by now, punt it, narrow down the results to 2 good onbes, take your expected value of .5, and walk. (I am an LIII candidate, but I just finished playing hockey so I am reading this board to get groggy!!)

gooman26 thank you so much your advice is a winner

sv102307 Wrote: ------------------------------------------------------- > To directly use the formula, i first just write > out the probabilities as below: > p(positive|user) = .99 > p(not positive| not user) = .99 > we need to calculate p(user|positive) > = (p(positive|user) * p(user)) / p(positive) > > p(positive) = p(positive| user) * p(user) + > p(positive|not user) *p(not user) > = .99 * .005 + .01 * * .995 > = .0149 > Correct. eitazul Wrote: ------------------------------------------------------- > using tree diagram > > > Drug User (.005) > ------ positive (.99) > ------ negative (.01) > > Non Drug User (.995) > ------ positive (.01) > ------ negative (.99) > > So P(Drug user|positive) = (.99 X .005) / {(.99 X > .005)(.01 X .995)} Not Correct.

gooman26, thanks for the advice. As far as solving problems, the tree diagram helps a lot.

Absolutely. The Bayes theorem looks extremely difficult and looks extremely impressive but if you do a tree diagram you’ll immediately see that it’s as easy as it can get. Do the tree diagram, then look at the formula again. It’ll make sense now.

Oppps…there’s a typo there…my bad. It should be a “+” sign instead . It should be this instead: P(Drug user|positive) = (.99 X .005) / {(.99 X .005) + (.01 X .995)} Note: The denominator is a total of positive users.