Loading...

​​Discussion 8B

​​Disjointness

​​Events ​​$A$​ and ​​$B$​ are disjoint if ​​$\Pr[A \cap B] = 0$​ (in the discrete case).

​​Independence

​​Events ​​$A$​ and ​​$B$​ are independent if

• ​​​​$\Pr[A \cap B] = \Pr[A] \Pr[B]$​

• ​​​​$\Pr[A] = \Pr[A \mid B]$​

​​

​​Special case to consider: if either event has a probability of ​​$0$​ of occurring, then are they independent? are they disjoint? What if either event has a probability of ​​$1$​ of occurring?

​​Union bound

​​Since ​​$\Pr[\bigcup A_i] = \sum \Pr[A_i] - \sum_{i,j} \Pr[A_i \cap A_j] + \cdots \pm \Pr[\bigcap A_i]$​ by inclusion-exclusion, we obtain the union bound ​​$\Pr[\bigcup A_i] \leq \sum \Pr[A_i]$​. The bound is tight when the events ​​$A_i$​ are disjoint (also known as mutually exclusive).

​​