Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I feel that

p(a,b) = the probability of event a and b happen in the same time.

p(a|b) = the probability of event a happens due to the event b happens.

For me, I think the meaning is quite the same. So what is the difference?

share|cite|improve this question
up vote 8 down vote accepted

I'm going to rephrase a little bit: $p(a,b)$ is the probability that both a and b happen. $p(a|b)$ is the probability that a happens, knowing that b has already happened.

I think the best way to think of these is to think of several examples.

Suppose we consider throwing 2 6-sided dice: suppose that condition 'A' is that the the numbers of the top faces of the two dice sum to 7, and 'B' is that die number 2 shows a 1.

Okay, now what is $p(a,b)$? Well, there is only 1 way in which this can happen: die 2 must show a 1, and the other a 6. As there are 36 possibilities that we all assume to have equal probability, $p(a,b) = 1/36$.

What is $p(a|b)$? So we know that die 2 is a 1. So the only way for the sum to be 7 is for die 1 to be a 6. As there are 6 possibilities for die 1, $p(a|b) = 1/6$.

Does that make sense?

Now, sometimes $p(a) = p(a|b)$, and this is when we call events a and b to be statistically independent.

share|cite|improve this answer
How about the relation between p(a,b) and p(a|b)? – A-letubby Jun 8 '11 at 8:57
is P(a,b) equals to P(a U b) ? – Samitha Chathuranga Sep 5 '15 at 6:25

$p(a|b)$ = the probability of event a happens given that the event b happens. The difference in words is critical. None of these have the sense of causation that due to implies. If b is unlikely, but a happens all the time b does, $p(a|b)$ can be quite high. If a is "I will be a millionaire tomorrow" and b is "I will win the lottery tonight", $p(a,b)$ is very low, but $p(a|b)$ is 1.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.