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We conclude this section on basic theorems by deriving a useful general expression for the
probability P(A ∩ B) that two events A and B both occur in the case where A (say) is the union
of a set of n mutually exclusive events A i . In this case

A ∩ B = (A 1 ∩ B) ∪ . . . ∪ (A n ∩ B),

wheretheeventsA i ∩B arealsomutuallyexclusive. Thus, fromtheadditionlaw(2.10)formutually
exclusive events, we find
∑
P(A ∩ B) = P(A i ∩ B). (2.15)
i

1.3. Conditional probability

Dependent Events. Conditional Probability. Multiplication Theorem of

Probabilities

Definition 3.1. Two events A and B are said to be independent if occurrence
of one is in no way affected by the occurrence or nonoccurrence of the other.
Otherwise event are called dependent.

So far we have defined only probabilities of the form ’what is the probability that event A
happens’? In this section we turn to conditional probability, the probability that a particular
event occurs given the occurrence of another, possibly related, event. For example, we may wish
to know the probability of event B, drawing an ace from a pack of cards from which one has
already been removed, given that event A, the card already removed was itself an ace, has
occurred. We denote this probability by P(B|A) and may obtain a formula for it by considering
the total probability P(A ∩ B) = P(B ∩ A) that both A and B will occur.

Example 3.1. An urn contains 5 white and 3 black balls. At random from an urn a
person draws a ball without returning it to the urn. The second ball is taken out
of the urn. What is the probability that this ball is black (event B).

Solution. The occurrence of the event B depends on of what color was the first ball. If the
first ball is white (event A), then the probability of the event B provided that the event A has
3
come is P A (B) = P(B|A) = .
7
If the first ball is black then the probability of the event B provided that the event A has not
2
come (event A has come) is P (B) = P(B|A) = .
A
7
P A (B) is called the conditional probability of the event B relative to the event A. In this case
the events A and B are dependent events.

Theorem 3.1. The probability of the product of two dependent events is equal to the
probability of one of them multiplied by the conditional probability of the other

P(AB) = P(A)P A (B) = P(B)P B (A). (3.1)
⋆

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