2.5.2 Independence of More Than Two Events

• The notion of independence of two events can be generalized to collections of more than two events.
• Although it is possible to extend the definition for two independent events by working in terms of conditional and unconditional probabilities,
• it is more direct and less cumbersome to proceed along the lines of the last proposition.

mutually independent

Events $A_1,\dots ,A_n$ are mutually independent if for

• every $k\left(k=2,3,\dots ,n\right)$
• every subset of indices $i_1,i_2,\dots ,i_k$ ,

$$\begin{array}{rl}& P\left(A_{i_1}\cap A_{i_2}\cap \dots \cap A_{i_k}\right)\\ =& P\left(A_{i_1}\right)\cdot P\left(A_{i_2}\right)\cdots \cdot P\left(A_{i_k}\right)\end{array}$$

• To paraphrase the definition, the events are mutually independent
  • if the probability of the intersection of any subset of the $n$ events is equal to the product of the individual probabilities.
• In using the multiplication property for more than two independent events, it is legitimate to replace one or more of the $A_i$‘s by their complements
  • e.g., if $A_1,A_2$ , and $A_3$ are independent events, so are $A_1^{\prime },A_2^{\prime }$ , and $A_3^{\prime }$ .
• As was the case with two events, we frequently specify at the outset of a problem the independence of certain events.
• The probability of an intersection can then be calculated via multiplication.

EX 2.36 solar cells