Independent Events

We define two events \(A\) and \(B\) to be independent in case:

\[\P(A\,B) = \P(A)\,\P(B)\]

and we often write \(A\perp B\) to denote independent events.

The meaning of what the indepence of events means is when we look at the conditional probability:

\[A\perp B \Longrightarrow \P(A\given B) = \P(A),\]

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 dices. The outcome of the first dice in no way influences the outcome of throwing with the second dice. 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 event ´Vase 1’ and ‘Red’ are evidently not independent.