2.4. Independent Events
We define two events \(A\) and \(B\) to be independent in case:
and we often write \(A\perp B\) to denote independent events.
When two events are independent it is simple to show that the conditional probability \(\P(A\given B)\) equals \(\P(A)\) i.e. knowledge about \(B\) does not influence the probability of \(A\). Evidently we also have \(A\perp B\Longrightarrow\P(B\given A)=\P(B)\) .
A well known example is throwing with two dice. The outcome of the first die in no way influences the outcome of throwing with the second die. Therefore throwing 2 times 6 in a row is \(1/6\times 1/6 = 1/36\) .
An example of dependent events can be found in the previous section about the marbles from the vases. The events ´Vase 1’ and ‘Red’ are evidently not independent.
Now consider events \(A\) and \(B\) and a third event \(C\).
Events \(A\) and \(B\) given \(C\) are conditional independent in case:
Independence of \(A\) and \(B\) does not imply conditional independence or vice versa.