Probability Theory, 2nd Edition

2 Conditional probabilities and independence

2.1 Conditional probabilities

In order to motivate the definition of conditional probabilities, let us start with the following easy example.

Example 2.1.1.

Roll a fair die twice. The probability of the event “sum of both rolls equals 5” is 1/9. Suppose now we were told that the first roll was an even number. Does this additional information make the event “sum equals 5” more likely? Or does it even diminish the probability of its occurrence? To answer this question, we apply the so-called technique of “restricting the sample space.” Since we know that the event $\mathit{B}=\{\text{First roll is even}\}$ had occurred, we may rule out elements in ${\mathit{B}}^{\mathit{c}}$ and restrict our sample space. Choose B as the new sample space. ...