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Statistical, mutual and pairwise independent events

Statistical, mutual and pairwise independent events

When an experiment consists of drawing objects at random from a given set of objects, it is termed
sampling a population. We need to distinguish between two different ways in which such a
sampling experiment may be performed. After an object has been drawn at random from the set
it may either be put aside or returned to the set before the next object is randomly drawn. The
former is termed ’sampling without replacement’, the latter ’sampling with replacement’.

Example 3.6. Find the probability of drawing two aces at random from a pack of
cards (i) when the first card drawn is replaced at random into the pack before
the second card is drawn, and (ii) when the first card is put aside after being
drawn.

Solution. Let A be the event that the first card is an ace, and B the event that the second card
is an ace. Now
P(A ∩ B) = P(A)P(B|A), (3.9)

and for both i and ii we know that P(A) = 4 = 1 .
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(i) If the first card is replaced in the pack before the next is drawn then P(B|A) = P(B) =
4 = 1 , since A and B are independent events. We then have
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1 1 1
P(A ∩ B) = P(A)P(B) = · = .
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(ii) If the first card is put aside and the second then drawn, A and B are not independent
and P(B|A) = 3 , with the result that
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1 3 1
P(A ∩ B) = P(A)P(B|A) = · = .
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Two events A and B are statistically independent if P(A|B) = P(A) (or equivalently if
P(B|A)= P(B)). In other words, the probability of A given B is then the same as the probability
of A regardless of whether B occurs. For example, if we throw a coin and a die at the same time,
we would normally expect that the probability of throwing a six was independent of whether a
head was thrown. If A and B are statistically independent then it follows that

P(A ∩ B) = P(A)P(B). (3.10)

In fact, on the basis of intuition and experience, (3.10) may be regarded as the definition of the
statistical independence of two events. The idea of statistical independence is easily extended to
an arbitrary number of events A 1 , A 2 , . . . , A n . The events are said to be(mutually) independent
if
P(A i ∩ A j ) = P(A i )P(A j ),

P(A i ∩ A j ∩ A k ) = P(A i )P(A j )P(A k ),

. . .

P(A 1 ∩ A 2 ∩ . . . ∩ A n ) = P(A 1 )P(A 2 ) . . . P(A n ),
for all combinations of indices i, j and k for which no two indices are the same. Even if all n events
are not mutually independent, any two events for which P(A i ∩ A j ) = P(A i )P(A j ) are said to be
pairwise independent.

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