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P(Red ball secondBlue ball first) =

Number of Red balls 5
=
Number of balls remaining after first pick 8

giving:

P(Blue ball first and Red ball second) = 4 × 5 = 5
9 8 18

10.4.1 Total Probability and Bayes Theorems
TOTAL PROBABILITY THEOREM
If [A , A , …, A ] is a partition of the total elementary occurrences set S, that is,
1
2
n
n
U A = S and A ∩ A = ∅ for i ≠ j
i
j
i
i=1
and B is an arbitrary event, then:

PB() = PB A P A( ) ( ) + P B A P A( ) ( ) +…+ PB A P A( ) ( ) (10.38)
1 1 2 2 n n

PROOF From the algebra of sets, and the definition of a partition, we can
write the following equalities:

B = B ∩ = B ∩ ( A ∪ A ∪…∪ A )
S
1 2 n
(10.39)
= ( B ∩ A ∪) ( B ∩ A ∪…∪) ( B ∩ A )
1 2 n
Since the events (B ∩ A ) and ( B ∩ A ) and are mutually exclusive for i ≠ j,
i j
then using the results of Pb. 10.7, we can deduce that:

PB() = PB( ∩ A ) + P B( ∩ A ) +…+ PB( ∩ A ) (10.40)
1 2 n

Now, using the conditional probability definition [Eq. (10.38)], Eq. (10.40) can
be written as:

PB() = PB A P A( ) ( ) + P B A P A( ) ( ) +…+ PB A P A( ) ( ) (10.41)
1 1 2 2 n n

This result is known as the Total Probability theorem.

© 2001 by CRC Press LLC