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Finally, we note that the concept of conditional probability may be straightforwardly extended to
several compound events. For example, in the case of three events A, B, C, we may write P(A ∩
B ∩ C) in several ways, e.g.

P(A ∩ B ∩ C) = P(C)P(A ∩ B|C) = P(B ∩ C)P(A|B ∩ C) =
= P(C)P(B|C)P(A|B ∩ C).

Theorem 3.2. Suppose {A i } is a set of mutually exclusive events that exhausts the
sample space S. If B and C are two other events in S, show that

∑
P(B|C) = P(A i |C)P(B|A i ∩ C).
i

PROOF. Using (3.2) and (2.15), we may write

∑
P(C)P(B|C) = P(B ∩ C) = P(A i ∩ B ∩ C). (3.13)
i
Each term in the sum on the RHS can be expanded as an appropriate product of conditional probabilities,

P(A i ∩ B ∩ C) = P(C)P(A i |C)P(B|A i ∩ C).

Substituting this form into (3.13) and dividing through by P(C) gives the required result. 2

1.4. Bernoulli trials and limit theorems

Bernoulli Trials

Suppose that we have an experiment such as tossing a coin or die repeatedly, choosing a marble
from an urn repeatedly, etc. Each toss or selection is called a trial. In any single trial there will be a
probability associated with a particular event such as head on the coin, 4 on the die or selection of
a red marble. In some cases this probability will not change from one trial to the next (as in tossing
a coin or die). Such trials are then said to be independent and are often called Bernoulli trials

after James Bernoulli who investigated them at the end of the 17-th century.
By Bernoulli trials we mean identical independent experiments in each of which an event A,
say, may occur with probability
p = P(A) (p ̸= 0)

or fail to occur with probability
q = 1 − p.
Occurrence of the event A is called a “success” and nonoccurrence of A (i.e., occurrence of the
complementary event A) is called “failure”.
Suppose that n independent trials are performed in every one of which the probability of the
event A is equal to p. It is required to find the probability P n (m) that the event A will appear m
times.

Example 4.1. A marksman fires 5 shots at a target. The probability of hitting
the target is one and the same p = 0.8 for all shots. Find the probability that
the target is hit one time, two times, …, five times. ,

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