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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?

2 Answers 2

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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.

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    How about the relation between p(a,b) and p(a|b)?2011-06-08
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    is P(a,b) equals to P(a U b) ?2015-09-05
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    does probability of P(a, b) = P(b, a) ??2018-01-02
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    Thanks for the clarification! Helped a lot. In other words, an intuitive way to think about this is the change that happens to the sample space as a result of our knowing B occurred. 36 outcomes -> 6 outcomes2018-09-18
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$p(a|b)$ = 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.