Independence of More Than 2 Events

Mutually Independent

Events $A_{1}, . . ., A_{k}$ are mutually independent if for every $k = 2, 3, . . ., n$ and every set of distinct indices $i_{1}, . . ., i_{k}$ :

P (A_{i_{1}} \cap A_{i_{2}} \cap \dots \cap A_{i_{k}}) = P (A_{i_{1}}) P (A_{i_{2}}) \dots P (A_{i_{k}})

Notice that we need every set of indices, so all possible combinations have to work here.

Quiz Example

A quiz in a newspaper contains a high-school, college, and graduate level question. Let $A_{1}$ denote the event that question 1 is answered correctly, and similarly for $A_{2}, A_{3}$ . Suppose that these events are independent and $P (A_{1}) = .8, P (A_{2}) = .5, P (A_{3}) = .1$ . Then the probability of answering all 3 is $.8 \cdot .5 \cdot .1 = .04$ . The chance that all three questions are answered wrong is $P (A_{1}^{'} \cap A_{2}^{'} \cap A_{3}^{'}) = (1 - .8) (1 - .5) (1 - .1) = .09$ .

For one question being right $P (A_{1} \cup A_{2} \cup A_{3}) = 1 - P (A_{1}^{'} \cap A_{2}^{'} \cap A_{3^{'}}) = 1 - .09 = .91$ .