# Conditional probability and joint probability

This is trivial,

Regarding probability and random variables. Are the following probabilities equivalent or similar?

$P(A=a \mid B=b)$ Conditional probability

$P(A=a,B=b)$ Joint probability

If not, what do they both mean?

• – georg Oct 24 '15 at 13:31
• @georg Exactly the clarity I was looking for! – Ben Winding Oct 24 '15 at 13:33

They are very different. The conditional probability is the probability that $A$ occurs given that you know that $B$ occurs. If, for example $A=B$ then this is $1$. The joint probability is the probability that both occur. If for example $A=B$ then this is $P(A)$.

To give a concrete example, consider one toss of a fair die. Let $A$ denote the event "you throw a $2$", let $B$ be the event "you throw less than a $4$.

Then the Conditional Probability $P(A|B)$ is the probability that you have thrown a $2$ Given that you know you have thrown less than a $4$. That value is $\frac 13$.

The joint Probability is the probability that both occur, which is $\frac 16$.

$P(A=a\mid B=b)$ is the condition probability that random variable $A$ has value $a$ when it is given that variable $B$ has value $b$

$P(A=a, B=b)$ is the joint probability that random variable $A$ has value $a$ and that random variable $B$ has value $b$

These are related as follows: $$P(A=a,B=b)= P(B=b)\;P(A=a\mid B=b)$$