THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1301
PROBABILITY AND STATISTICS I
EXAMPLE CLASS 2
Review
Definition
For any two events A and B, the conditional probability of A given the occurrence of B is written as
P (A|B) and is defined as
P (A ∩ B)
P (A|B) =
P (B)
provided that P (B > 0).
Multiplication theorem
For any two events A and B with P (B) > 0,
P (A ∩ B) = P (B)P (A|B).
For any three events A, B, C with P (B ∩ C) > 0,
P (A ∩ B ∩ C) = P (C)P (B|C)P (A|B ∩ C).
Independence
Two events A and B are called independent if and only if
P (A ∩ B) = P (A)P (B).
If P (A) > 0,then A and B are independent if P (B|A) = P (B).
Bayes’ Theorem (Bayes’ rule, Bayes’ law)
For any two event A and B with P (A) > 0 and P (B) > 0,
P (B|A) = P (A|B)
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P (B)
.
P (A)
Bayes’ Theorem
If B1 , B2 , ..., Bk are mutually exclusive and exhaustive events (i.e. a partition of the sample space), and
A is any event with P (A) > 0, then for any Bj ,
P (Bj |A) =
P (A|Bj )P (Bj )
P (Bj )P (A|Bj )
= k
,
P (A)
X
P (Bi )P (A|Bi )
i=1
where k can also be ∞.
Law of total probability
If 0 < P (B) < 1, then P (A) = P (A|B)P (B) + P (A|B c )P (B c ) for any A.
If B1 , B2 , ..., Bk are mutually exclusive and exhaustive events (i.e. a partition of the sample space), then
for any event A,
k
X
P (A) =
P (A|Bj )P (Bj ),
j=1
where k can also be ∞.
Problems
Problem 1
A and B are two events. Suppose that P (A|B) = 0.6, P (B|A) = 0.3, and P (A∪B) = 0.72. LetP (A) = a.
(a) Express P (A ∩ B) and P (B) in terms of a.
(b) Using the results of (a), or otherwise, find the value of a.
(c) Are A and B independent events? Explain your answer briefly.
Problem 2
A and B are two events. Suppose that P (B c |A) = 43 , P (Ac |B) = 35 , and P (Ac ) = 52 , where Ac and B c
are complementary events of A and B respectively. Let P (B) = p, where 0 < p < 1.
(a) Find P (A ∩ B c ).
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(b) Express P (Ac ∩ B) in terms of p.
(c) Using the fact that Ac ∪ B is the complementary event of A ∩ B c , or otherwise, find the value of p.
(d) Are A and B mutually exclusive? Explain your answer.
Problem 3
(a) Prove
(C1 ∪ C2 ∪ · · · ∪ Ck )c = C1c ∩ C2c ∩ · · · ∩ Ckc
(b) Say that C1 , C2 , · · ·, Ck are independent events that have respective probabilities p1 , p2 , · · ·, pk .
Argue that the probability of at least one of C1 , C2 , · · ·, Ck is equal to
1 − (1 − p1 )(1 − p2 ) · · · (1 − pk )
(c) Let C1 , C2 , C3 be independent events with probabilities 21 , 13 , 41 , respectively. Compute
(i) P (C1 ∪ C2 ∪ C3 )
(ii) P [(C1c ∩ C2c ) ∪ C3 ]
Problem 4
A small plane have gone down, and the search is organized into three regions. Starting with the likeliest,
they are:
Region
Initial Chance Plane is There Chance of Being Overlooked in the Search
Mountains
0.5
0.3
Praire
0.3
0.2
Sea
0.2
0.9
The last column gives the chance that if the plane is there, it will not be found, it will not be found. For
example, if it went down at sea, there is 90% chance it will have disappeared, or otherwise not be found.
Since the pilot is not equipped to long survive a crash in the mountains, it is particularly important to
determine the chance that the plane went down in the mountains.
(a) Before any search is started, what is this chance?
(b) The initial search was in the mountains, and the plane was not found. Now what is the chance
the plane is nevertheless in the mountains?
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(c) The search was continued over the other two regions, and unfortunately the plane was not found
anywhere. Finally now what is the chance that the plane is in the mountains?
(d) Describing how and why the chances changed from (a) to (b) to (c).
Problem 5
Let S = {1, 2, · · ·, n} and suppose that A and B are, independently, equally likely to be any of the 2n
subsets ( including the null set and S itself) of S.
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(a) Show that P (A ⊂ B) = ( )n .
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HINT: Let N (B) denote the number of elements in B. Use
P (A ⊂ B) =
n
X
P (A ⊂ B|N (B) = i)P (N (B) = i)
i=0
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(b) Show that P (A ∩ B = φ) = ( )n .
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