Chapter 2, Section 7B Independence

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Chapter 2, Section 7BIndependence
? John J Currano, 09/17/2007
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(Stochastically) Independent Events Intuitively,
two events A and B are independent events if
knowledge of the occurrence of one of them does
not change the probability that the other occurs.
Formally, we define two events A and B to be
independent if any of the following is true
Two events which are not independent are called
dependent.
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IndependenceP(A?B) P(A) P(B)
 
Example. A bowl contains 7 blue chips and 3 red
chips. Two chips are drawn at random, in order
and with replacement. Let S set of all ordered
pairs of distinct chips, A be the event that
the first chip drawn is red, and B be the event
that the second chip drawn is blue. Are the
events A and B independent? Solution 1 A ? B
event first is red and second is blue.
3 ? 10 30
3 ? 7 21
10 ? 10 100
S A B A ? B
10 ? 7 70
P(A) P(B) P(A ? B) P(A) P(B)
P(A) 30/100 0.3 P(B) 70/100 0.7 P(A ? B)
21/100 0.21 P(A) P(B) (0.3)( 0.7) 0.21
P(A) 30/100 0.3 P(B) 70/100 0.7 P(A ? B)
21/100 0.21 P(A) P(B)
P(A) 30/100 0.3 P(B) 70/100 0.7 P(A ? B)
P(A) P(B)
P(A) 30/100 0.3 P(B) P(A ? B) P(A) P(B)
Since P(A?B) 0.21 P(A) P(B), A and B are
independent.
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IndependenceP(B A) P(B)
 
Example. A bowl contains 7 blue chips and 3 red
chips. Two chips are drawn at random, in order
and with replacement. Let S set of all ordered
pairs of distinct chips, A be the event that
the first chip drawn is red, and B be the event
that the second chip drawn is blue. Are the
events A and B independent? Solution 2 P(B)
(10?7) /( 10?10) 7/10 as before, and
P(BA)
7/10 since there will be
7 blue chips and 3 red chips in the bowl when
the second chip is drawn. Thus, P(B A)
P(B), so A and B are independent. Intuitively,
since the first chip is replaced before the
second chip is drawn, there will be 10 chips (7
blue) in the bowl when the second chip is drawn,
regardless of the color of the first chip drawn.
Thus, the probability that B occurs (second is
blue) is not affected by the occurrence (or
non-occurrence) of A (first is red).
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IndependenceP(B A) P(B)
 
Example. A bowl contains 7 blue chips and 3 red
chips. Two chips are drawn at random, in order
and without replacement. Let S set of all
ordered pairs of distinct chips, A be the event
that the first chip drawn is red, and B be the
event that the second chip drawn is blue. Are the
events A and B independent? Solution.
10 ? 9 90
S B
63 / 90 7/10
P(B)
9 ? 7 63
_ ? 7
P(BA) 7/9, since given that A occurred, there
are 9 chips (7 blue) in the bowl when the second
chip is drawn.
P(BA)
Since P(BA) 7/9 ?? 7/10 P(B), A and B are
dependent.
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• Independent Events vs. Mutually Exclusive Events
• Be careful not to confuse these two terms.
• Two events A and B cannot be both independent and
  mutually exclusive unless one or both has
  probability 0.
• If both P(A) ? 0 and P(B) ? 0, then P(A) ? P(B)
  ? 0.
• If A and B are independent and both P(A) ? 0
  and P(B) ? 0,
• then P(A ? B) P(A) ? P(B) ? 0.
• If A and B are mutually exclusive,
• then P(A ? B) P(?) 0. P(?) 0 is
  shown in 2.8

Intuitively, if P(A) ? 0, P(B) ? 0, and A and B
are mutually exclusive, then the occurrence of
one (A, say) precludes the occurrence of the
other (B), which changes the probability of B
(from nonzero to zero). Thus A and B cannot be
independent.
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Example. An urn contains four balls One is drawn
at random. Let A , B
, C . Show 1. P(A) P(B) P(C)
1/2 2. P(A ? B) P(A ? C) P(B ? C)
1/4 3. A, B, and C are pairwise independent. 4.
P(A ? B ? C) 1/4 5. P(A ? B ? C) 1/4 ? 1/8
P(A) ? P(B) ? P(C) Definition. Events A, B, and C
are mutually independent iff 1. They are
pairwise independent 2. P(A ? B ? C) P(A)
? P(B) ? P(C). This definition is extended to
mutual independence of four or more events,
requiring the product rule to hold for all pairs,
triples, etc.
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• Suppose that A and B are events in a random
  experiment with P(B) gt 0. Prove
• If B ? A, then
• If A ? B, then
• If A and B are disjoint, then

• Key Points
• If B ? A, then A ? B B.
• If A ? B, then A ? B A.
• If A and B are disjoint, then A ? B ?.