1 The theorem of total probability

To establish this result we start with the definition of a partition of a sample space.

1.1 A partition of a sample space

The collection of events A 1 , A 2 , A n is said to partition a sample space S if

  1. A 1 A 2 A n = S
  2. A i A j = for all i , j
  3. A i for all i

In essence, a partition is a collection of non-empty, non-overlapping subsets of a sample space whose union is the sample space itself. The definition is illustrated by Figure 10.

Figure 10

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If B is any event within S then we can express B as the union of subsets:

B = ( B A 1 ) ( B A 2 ) ( B A n )

The definition is illustrated in Figure 11 in which an event B in S is represented by the shaded region.

Figure 11

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The bracketed events ( B A 1 ) , ( B A 2 ) ( B A n ) are mutually exclusive (if one occurs then none of the others can occur) and so, using the addition law of probability for mutually exclusive events:

P ( B ) = P ( B A 1 ) + P ( B A 2 ) + + P ( B A n )

Each of the probabilities on the right-hand side may be expressed in terms of conditional probabilities:

P ( B A i ) = P ( B | A i ) P ( A i ) for all i

Using these in the expression for P ( B ) , above, gives:

P ( B ) = P ( B | A 1 ) P ( A 1 ) + P ( B | A 2 ) P ( A 2 ) + + P ( B | A n ) P ( A n ) = i = 1 n P ( B | A i ) P ( A i )

This is the theorem of Total Probability. A related theorem with many applications in statistics can be deduced from this, known as Bayes’ theorem.