Partition of the sample space, and total probability

The set of events ${A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}}$ is a Partition of the Sample Space $Ω$ iff:

$A_{i} \cap A_{j} = \emptyset, \forall i \neq j;$ all the events in the partition are mutually exclusive.
$\cup_{i = 1}^{n} A_{i} = Ω$
$P (A_{i}) > 0, \forall i$

Given a partition ${A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}}$ , it is possible to calculate the probability of any event $B$ defined in the sample space $Ω$ , applying the Total Probability theorem, defined with the following expression:

P (B) = \sum_{i = 1}^{n} P (B / A_{i}) P (A_{i})

And applying the Bayes theorem: $P (\frac{A}{B}) = \frac{P (\frac{B}{A}) P (A)}{\sum_{i = 1}^{n} P ({\frac{B}{A}}_{i}) P (A_{i})}$

Example: A transmission system can send one of the following letters in binary code: A=100, B=110, C=111. The prior probability of transmitting each letter is: P(A)=0.4; P(B)=0.4; P(C)=0.2.

Randomly select one symbol of the transmitted signal. Calculate the probability of this symbol to be 1.

Answer: $P (1) = P (\frac{1}{A}) P (A) + P (\frac{1}{B}) P (B) + P (\frac{1}{C}) P (C) = 0.6$
Suppose the selected symbol is 1. Calculate the most likely transmitted letter.