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F7 Mathematics and Statistics
Chapter 13 Probability: Probabilities of Compound Events
F7-MS-Ch13-1
Section 13.1THE ADDITION RULE(加法原則)
Section 13.1.1
The General Addition Rule for two events(兩件事件的情況)
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) =
2
4
19
, P(B) =
and P(AB)=
. Find
5
5
30
P(AB).
C.W.
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?
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)
Example
[see p218-219 eg 13.4]
Section 13.1.2 The Special Addition Rule for Mutually Exclusive 互斥 Events
If A1, A2, …, Ak are mutually exclusive, then
P(A1A2…Ak) = P(A1) + P(A2) +… + P(Ak).
Example 2.
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.
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F7 Mathematics and Statistics
Chapter 13 Probability: Probabilities of Compound Events
F7-MS-Ch13-2
Example 3.
A carton contains 12 eggs of which 3 are rotten If 3 eggs are randomly selected
from the carton without replacement, what is the probability that at least 2 of them
are rotten?
Section 13.2 Conditional Probabilities(條件概率)
P(A|B) -------probability of A, given that B has already occurred
P( A  B)
P(A|B) =
P( B)
Example 4.
Given that a heart is picked at random from a pack of 52 playing cards, find the
probability that it is a picture card.
Example 5.
When a die is thrown, an odd number occurs. What is the probability that the number is
prime?
Example 6.
Two tetrahedral, with faces labeled 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.
H.W. EX13.2 (Q4,5,6,10)
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F7 Mathematics and Statistics
Chapter 13 Probability: Probabilities of Compound Events
F7-MS-Ch13-3
Section 13.3 The Multiplication Rule
Section 13.3.1 The General Multiplication Rule
The general multiplication rule for events A and B in the sample space S:
P(AB) = P(A) P(B|A)
Or,
P(AB) = P(B) P(A|B)
Example7
A carton contains 12 eggs of which 3 are rotten If 3 eggs are randomly selected from the
carton without replacement,
(a) both are rotten ?
(b) exactly one is rotten ?
(c) at least one is rotten ?
**
Example [see p.234 e.g. 13.9]
P(ABC) = P(A) P(B|A) P(C|AB)
H.W. EX13.3 (Q1,2,4,6,10,11)
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F7 Mathematics and Statistics
Chapter 13 Probability: Probabilities of Compound Events
F7-MS-Ch13-4
Section 13.4 Independent Events and the Special Multiplication Rule
Section 13.4.1
Independence of Two Events
Event B is independent of event A , P(B|A) = P(B).
Two events A and B are independent iff P(AB) = P(A)P(B)
Example 8
Two events A and B are such that P(A) =
2
3
1
, P(B) =
and P(A  B) =.
4
4
2
Are A and B independent events?
Example 9
Women(W)
Men(M)
Favours(F)
56
84
Oppose(O)
24
36
Suppose a person is selected at random from the sample.
(a) What is P(F), the prob. that the person is in favour of the proposal?
(b) What is P(F|W) ?
(c) Are F and W independent ?
H.W. EX.13.4 (Q3,5,10,12,14,17)
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Chapter 13 Probability: Probabilities of Compound Events
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C.W
Conditional Probability
1)Suppose a box contains 3 white balls and 5 red balls.
(i) Balls are drawn randomly one by one without replacement(抽出後不再放) from it. What
is the probability that the second ball drawn will be red, given that the first ball drawn is
white? (試求第 2 次抽出的球為紅球的概率)
(ii) 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.
現隨機地由盒內抽出 1 球，抽出後再放回盒內。若首 2 次抽出的球皆為白色，試求
第 3 次抽出的球為白球的概率。
2)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. 一信用咭公司調查大學生的新戶口情況。
若抽取 160 名學生為樣本，查詢他們是否擁有信用咭 X 或信用咭 Y，結果如下：
credit card X
credit card Y
Yes
No
Yes
50
20
No
30
60
Let event A = students possessed two credit cards, event B = students possessed at least one
credit card.,event C = students did not possess any card., event D = students possessed a credit
card X., event E = students possessed a credit card Y.
設 事件 A = 擁有兩款信用咭的學生， 事件 B = 最少擁有一款信用咭的學生， 事
件 C = 沒有任何信用咭的學生，事件 D = 擁有信用咭 X 的學生，事件 E = 擁有信用咭
Y 的學生。
Find the probabilities of each of these events A,B,C,D,E, A  B, A  B . Find also P( A E ) ,
P( A D) .
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F7 Mathematics and Statistics
Chapter 13 Probability: Probabilities of Compound Events
Section 13.5
F7-MS-Ch13-6
Bayes’ Theorem 貝葉斯定理
Section 13.5.1 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( E ) P( A | E )
i 1
i
i
Example [see p.204]
Section 13.5.2
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 )
P(Er |A) = k
for r =1,2,…,k
 P( E j ) P( A | E j )
j 1
[Refer to your textbook]
C.W.
1) Suppose there are three identical boxes which contain different number of white and
black balls.
1 st box
2 nd box
3 rd box
Number of white balls
8
6
4
Number of black balls
3
5
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. (若抽出的球為白球，求它是從第 1 個盒子抽出來的概率。)
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F7 Mathematics and Statistics
Chapter 13 Probability: Probabilities of Compound Events
F7-MS-Ch13-7
2) The marketing manager 市埸經理 of a soft drink manufacturing firm is planning to
introduce a new band 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. (市埸經理希望找出，若新品種可樂已得到一受歡迎的
報告，它成功推廣的概率為何？)
3)
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
MTR
5
8
Bus
2
8
Tram
1
8
Probability of being late
1
4
5
9
7
8
(a) He was late for his visit. Find the probability that he had travelled by MTR.
已知他遲到，求他乘搭地鐵的概率。
(b) He was not late for his visit. Find the probability that he had travelled by Bus.
已知他沒有遲到，求他乘搭公共巴士的概率。
EX13.5 (Q3,5,10)
H.W. (Q1-8), 9,10
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