We define conditional probability
\(P(E_i|E_j):=\dfrac{P(E_i\land E_j)}{P(E_j)}\)
We can show this is between \(0\) and \(1\).
\(P(E_j)=P(E_i\land E_j)+P(\bar{E_i}\land E_j)\)
\(P(E_i|E_j):=\dfrac{P(E_i\land E_j)}{ P(E_i\land E_j)+P(\bar{E_i}\land E_j)}\)
We know:
\(P(E_i|E_j):=\dfrac{P(E_i \land E_j)}{P(E_j)}\)
\(P(E_j|E_i):=\dfrac{P(E_i \land E_j)}{P(E_i)}\)
So:
\(P(E_i|E_j)P(E_j)=P(E_j|E_i) P(E_i)\)
\(P(E_i|E_j)=\dfrac{P(E_j|E_i) P(E_i)}{P(E_j)}\)
Note that this is undefined when \(P(E_j)=0\)
Note that for the same event,
\(P(E_i|E_j)=\dfrac{P(E_i\land E_j)}{P(E_j)}\)
\(P(E_i|E_j)=0\)
For the same outcome:
\(P(E_i|E_i)=\dfrac{P(E_i\land E_i)}{P(E_i)}\)
\(P(E_i|E_i)=\dfrac{P(E_i)}{P(E_i)}\)
\(P(E_i|E_i)=1\)
From the definition of conditional probability we know that:
\(P(E_i|E_j):=\dfrac{P(E_i\land E_j)}{P(E_j)}\)
\(P(E_j|E_i):=\dfrac{P(E_i\land E_j)}{P(E_i)}\)
So:
\(P(E_i\land E_j)=P(E_i|E_j)P(E_j)\)
\(P(E_i\land E_j)=P(E_j|E_i)P(E_i)\)
So:
\(P(E_i|E_j)P(E_j)=P(E_j|E_i)P(E_i)\)
Events are independent if:
\(P(E_i|E_j)=P(E_i)\)
Note that:
\(P(E_i\land E_j)=P(E_i|E_j)P(E_j)\)
And so for independent events:
\(P(E_i\land E_j)=P(E_i)P(E_j)\)