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Assuming $X_i$ is a random variable and $P$ is a probability measure, often I read these two notations in books:

  • $P(X_2 =1|X_1=3)$
  • $P(X_2 = 1,X_1=3)$

Sometimes even $P(X_2 = 1 \wedge X_1=3)$. Are they all equivalent or do I confuse something here? Thanks for any suggestions or corrections!

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For the first, see conditional probability – user12205 Dec 4 '11 at 12:44
up vote 6 down vote accepted

$P(X_2=1 | X_1=3)$ is the probability that $X_2=1$ given that $X_1=3$. This is called a conditional probability. You assume that $X_1=3$, and from there, compute the probability that $X_2=1$.

$P(X_2=1 , X_1=3)$ is the probability that $X_2=1$ and $X_1=3$. Here, you are computing the probability that both of the events occur simultaneously.

They are quite different:


Toss a coin twice in succession. The probability that the first flip is a head and the second flip is a head is $1/4$. But, the probability that the second flip is a head, given that the first flip is a head, is 1/2.

But they are related: $$ \tag{1}P(X_2=1 | X_1=3)={P(X_2=1 , X_1=3)\over P(X_1=3)}, \text{ if } P(X_1=3)\ne0. $$

The notation $P(X_2 = 1 \wedge X_1=3)$ is just a different way of writing $P(X_2 = 1 , X_1=3)$.


A word about formula (1) (warning: I'm hand-waving here). Given that the event $B$ happens, your sample space is "reduced" to those outcomes and only those outcomes that are in $B$. Then, within this reduced sample space, the probability of the event $A$ is a certain proportion of the reduced sample space. What is the proportion? Well, it's the the one given by formula (1).

This may be seen more easily by considering a Venn diagram, where probabilities are interpreted as areas. $P(A|B)$ is the ratio of the area of $A\cap B$ with the area of $B$.

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