Download 2.1 Discrete and Continuous Variables

Chapter 4 Probability: Probabilities of
Compound Events
4.1THE ADDITION RULE
4.1.1 The General Addition Rule
4.1.2The Special Addition Rule for Mutually Exclusive Events
4.2 Conditional Probabilities
4.3 The Multiplication Rule
4.4 Independent Events and the Special Multiplication Rule
4.4.1Independence of Two Events
4.4.2Independence of More Than Two Events and the Special
Multiplication Rule
4.5 Bayes’ Theorem
4.5.1 The Total Probability
4.5.2 Bayes’ Theorem
4.1THE ADDITION RULE
 4.1.1 The General Addition Rule
The general addition rule for two events, A and
B, in the sample space S:
P(AB) = P(A) + P(B) – P(AB)
Example 1
 Events A and B are such that P(A) =19/30 ,
P(B) =2/5 and P(AB)=4/5 . Find P(AB).
(Ans: 9/30)
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Example 2
In a group of 20 adults, 4 out of the 7 women and 2 out
of the 13 men wear glasses. What is the probability
that a person chosen at random from the group is a
woman or someone who wears glasses? (Ans: 1/5)
Example 3
A class contains 10men and 20 women of which half
the men and half the women have brown eyes. Find
the probability p that a person chosen at random is a
man or has brown eyes. (Ans: 2/3)
 The General Addition Rule for Three Events
P(ABC) = P(A) + P(B) + P(C) – P(AB) – P(AC) – P(BC)
+ P(ABC)
4.1.2 The Special Addition Rule for Mutually
Events
Exclusive
If A1, A2, …, Ak are mutually exclusive, then
P(A1A2…Ak) = P(A1) + P(A2) +… + P(Ak).
 Example 1
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Records in a music shop are classed in the following sections:
classical, popular, rock, folk and jazz. The respective probabilities
that a customer buying a record will choose from each section are
0.3, 0.4, 0.2, 0.05 and 0.05. Find the probability that a person (a)
will choose a record from the classical or the folk or the jazz
sections, (b) will not choose a record from the rock or folk or
classical sections.
4.2 Conditional Probabilities
If A and B are two events and P(A)  0 and P(B)  0, then
the probability of A, given that B has already occurred is
written P(A|B) and
P(A|B) = P( A  B)
P( B)
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Example 1
Given that a heart is picked at random from a
pack of 52 playing cards, find the probability
that it is a picture card.
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Example
When a die is thrown, an odd number occurs. What is
the probability that the number is prime?
Example
Two tetrahedral, with faces labelled 1,2,3 and 4, are
thrown and the number on which each lands is noted.
The ‘score’ is the sum of these two numbers. Find the
probability that the score is even, given that at least
one die lands on a 3.
4.3 The Multiplication Rule
The general multiplication rule for events A and B in
the sample space S:
P(AB) = P(A) P(B|A)
P(AB) = P(B) P(A|B)
P(ABC) = P(A) P(B|A) P(C|AB)
4.4 Independent Events and the
Special Multiplication Rule
 4.4.1
Independence of Two Events
If the occurrence or non-occurrence of an event A does not
influence in any way the probability of an event B, then
event B is independent of event A and P(B|A) = P(B).
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Two events A and B are independent iff P(AB) =
P(A)P(B)
Note: If two evens are mutually exclusive, then P(AB) =
_______. So for two events to be both independent and
mutually exclusive we must have P(A) P(B) = P(AB) =
________. This is possible only if either P(A) = _________
or P(B) = __________.
 Example 1
 A die is thrown twice. Find the probability of obtaining a 4 on the first
throw and an odd number on the second throw.
 Example 2
 A bag contains 5 red counters and 7 black counters. A counter is
drawn from the bag, the colour is noted and the counter is replaced.
A second counter is then drawn. Find the probability that the first
counter is red and the second counter is black.
 Example 3
 A fair die is thrown twice. Find the probability that (a) neither throw
results in a 4, (b) at least one throw results in a 4.
 Example 4
 Two events A and B are such that P(A) = , P(A|B) = and P(B|A) = .
 Are A and B independent events? (b) Are A and B mutually exclusive
events?
 (c) Find P(AB). (d) Find P(B).
 4.4.2 Independence of More Than Two Events and the
Special Multiplication Rule
If k events A1, A2,…., Ak are independent, then
P(A1  A2 …. Ak) = P(A1)P(A2)…P(Ak)
 Example 1
 A die is thrown four times. Find the probability that a
5 is obtained each time.
 Example 14
 Three men in an office decide to enter a marathon
race. The respective probabilities that they will
complete the marathon are 0.9, 0.7 and 0.6. Find the
probability that at least two will complete the
marathon. Assume that the performance of each is
independent of the performances of the others.
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C.W
Conditional Probability
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1)
In a family of two children with at least one girl. What is
the probability that the other one is a boy?
2)
Suppose a box contains 3 white balls and 5 red balls.
Balls are drawn randomly one by one without
replacement from it. What is the probability that the
second ball drawn will be red, given that the first ball
drawn is white?
Balls are drawn randomly one by one with replacement
from it. What is the probability that the third ball drawn
will be white, given that the first two balls drawn are
white.
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 3)
 A credit card company has surveyed new
accounts from university students. Suppose a
samples of 160 students indicated the following
information in terms of whether the student
possessed a credit card X and/or a credit card Y.
credit card X
credit card Y
Yes
No
Yes
50
20
No
30
60
 4.) Let event A = students possessed two credit cards.
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event B = students possessed
at least one credit
A  B, A  B
card.
. Find also
event C = students did
not possess any card.
,
event D = students possessed
a credit card X.
.
event E = students possessed a credit card Y.
Find the probabilities of each of these events A,B,C,D,E,
Find also P( A E ) and . P( A D)
 5) A fair coin is tossed three times
 Let event A = Head appears on first toss.
 event B = Head appears on second toss.
 event C = Head appears on all three tosses.
 To find whether A and B, B and C, C and A
are independent.
4.5 Bayes’ Theorem
 4.5.1 The Total Probability
Suppose a sample space S is partitioned into k mutually
exclusive events Ej (j = 1,2,…,k), i.e. S = E1E2….Ek
with EiEj =  for ij, then
P(A) = P(E1)P(A|E1) + P(E2)P(A|E2) +…+ P(Ek)P(A|Ek)
k
=
 P( Ei ) P( A | Ei )
i 1
 4.5.2
Bayes’ Theorem
Let the sample space S be partitioned into mutually
exclusive events Ej’s (j = 1,2,…,k) and let A be an
event in S. Then the probability of Er conditional on A
is
P ( E r ) P ( A | E r ) for r =1,2,…,k
P(Er |A) =
k
 P( E
j 1
j
) P( A | E j )
1)
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Suppose there are three identical boxes which contain different number
of white and black balls.
Number of white balls
Number of black balls
1 st box
8
3
2 nd box
6
5
3 rd box
4
7
A box is selected at random and a ball is drawn from it randomly .
(I) What is the probability that a white ball is chosen?
(ii) Suppose a white ball is chosen, find the probability that this white ball
comes from the 1st box.
 2) The marketing manager of a soft drink
manufacturing firm is planning to introduce a new
rand of Coke into the market. In the past, 30 % of the
Coke introduced by the company have been
successful, and 70% have not been successful.
Before the Coke is actually marketed, market
research is conducted and a report, either favorable
or unfavourable, is compiled. In the past, 80% of the
successful Coke received favourable reports and
40% of the unsuccessful Coke also received
favourable reports. The marketing manager would
like to know the probability that the new brand of
Coke will be successful if it receives a favourable
report.
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3)
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A man decided to visit his friend at North Point. He can
reach there by MTR, Bus or Tram respectively. The
following information is given:
Probability of being taken
Probability of being late
MTR
5/8
1/4
Bus
2/8
5/9
Tram
1/8
7/8
(i) He was late for his visit. Find the probability that he had
travelled by MTR.
(ii) He was not late for his visit. Find the probability that he
had travelled by Bus.