2.12 TOTAL PROBABILITY
The law of total probability is an extension of Bayes' formula to a partition of events in the sample space Ω. It is a convenient means for computing the probability of an event in terms of conditional probabilities that might be easier to compute for some problems, such as the BSC discussed in Example 2.43.
Theorem 2.3 (Total probability). Let {Bn}, 
, be a partition of Ω. Then
(2.76) 
Proof. Since {Bn} form a partition, 
and 
for 
. We also have
(2.77) 
so that
(2.78) 
Applying conditional probability to each term in the sum gives P(ABn) = P(A|Bn)P(Bn), which completes the proof.
Example 2.44. Continuing with the BSC in Example 2.43 with input X and output Y, (2.75) was generated by conditioning on a partition for X; the results for each value of Y are
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