Lesson 1

Events \(A\) and \(B\) are called independent if \[\Pr[A \cap B] = \Pr[A] \cdot \Pr[B] \ .\] If the probabilities of events \(A\) and \(B\) are positive, this is equivalent to the conditions \(\Pr[A \mid B] = \Pr[A]\) and \(\Pr[B \mid A] = \Pr[B]\): the occurrence of event \(B\) does not change the probability of \(A\). An event \(B\) with zero probability is considered independent of any other event (including itself). Independence is often also an assumption, rather than a property: when rolling a die/coin a second time, we naturally assume that the result does not depend on the outcome of the first roll.