F.3 Mathematics Supplementary Notes
Chapter 12 Introduction to Probability
Chapter 12 Introduction to Probability
Important Terms
theoretical probability
experimental probability
geometric probability
random
expected value
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Name:______________(
Class: F.3 (
sample space
outcome
tree diagram
impossible event
certain event
Revision Notes
1. Probability
(a) Probability is a number used to indicate the likelihood of an event happening.
(b) Equally likely outcomes are those having the same chance of happening.
(c) Let E stand for an event and P(E) stand for its probability. If all outcomes are equally likely
Number of outcomes favourable of E
outcomes, then P( E ) 
.
Total number of possible outcomes
(d) For any event E, its probability P(E) has the following properties:
(i) 0  P( E )  1 .
(ii) If P(E) is large, the chance that E happens is larger.
(iii) If E' stands for the event "E does not happen", then P(E) + P(E') =1.
2.
The following methods help to list out all the possible outcomes of a situation systematically.
(a) Tree diagram
(b) Tabulation
3.
The probability obtained by considering measures of geometric figures, such as lengths or areas, is
called a geometric probability.
4.
Experimental Probability
(a) A probability found by deductive reasoning is called a theoretical probability; a probability
based on relative frequencies found from empirical sources is called an experimental probability.
(b) The outcomes of a situation may not be equally likely outcomes.
(c) For a large number of trials, experimental probability  theoretical probability.
5.
Expected Number of Occurrences
Suppose in a trial, the probability of an event is p. After n trials, we expect this event will
occur np times.
6.
Expected value of an experiment
Suppose a trial has n possible outcomes, and the probabilities of the outcomes are p1, p2…… pn,
respectively. If the values obtained after the occurrence of these n outcomes are respectively a1,
a2…… an, then the expected value of this trial is a1p1 + a2p2+……+ an pn .
)
)
F.3 Mathematics Supplementary Notes
Chapter 12 Introduction to Probability
Exercise
1. A die is thrown 120 times and the results are
3.
recorded as follows:
1
2
Number
24
Frequency 28
(a) Find the value of y.
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A box contains 18 apples where 3 of them are
bad. What is the probability of picking a
good apple at random?
3
16
4
22
5
y
6
18
(b) Find the experimental probability of getting
(i) a ‘5’;
4.
A multiple choice question has 4 options of
which only one is correct. For that question,
what is the probability of choosing the
wrong answer at random.
(ii) an odd number;
(iii) a number greater than 4.
2.
In tossing a fair die, what is the probability
of getting a number which is
(a) not greater than 6?
5.
In F.3A, the number of boys is 15 more than
that of girls. If the probability of selecting a
girl at random is
there in the class?
(b) less than 3?
2
, how many boys are
9
F.3 Mathematics Supplementary Notes
Chapter 12 Introduction to Probability
6. A bag contains 3 white balls, 7 black balls.
(a) If a ball is drawn at random from the bag,
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(c) Find the value of n.
what is the probability of getting a white ball?
(b) If a black ball is drawn without
replacement(不放回原袋), what is the probability
of getting another black ball from the bag?
8. A card is drawn at random from a pack of
52 playing cards (with no Jokers). What is the
probability of getting
(a) a spade(黑桃) King?
(b) a Jack?
7. A bag contains 12 white balls, 3 red balls
and n black balls. All the balls are identical. If a
ball is drawn at random, the probability of
drawing a white ball and a red ball are
(c) a diamond(方塊)Queen of a heart(紅心)Ace?
2
1
and
5
10
respectively.
(a) Find the probability that the ball drawn
is not white.
9.
(b) Find the probability that the ball drawn
is black.
A box contains 12 cards which are marked
with a number from 1 to 12 on each card
respectively. If a card is drawn at random what is
the probability of getting a number which is
(a) a multiple(倍數) of 3?
F.3 Mathematics Supplementary Notes
Chapter 12 Introduction to Probability
(b) a multiple of 4?
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(iii) at least one white ball,
(iv) the balls with the same color,
(c) a multiple of “3 or 4”?
(v) no white ball.
11. The weather may be rainy, cloudy or fine
in combination(組合) with being cold, warm or
10. There are two boxes. Each box contains 1
red ball(R), 1 white ball(W) and 1 blue ball(B). If
hot.
(a) Complete the following tree diagram to show
a ball is drawn from each box at random,
(a) list all possible outcomes in the
following table
all possible outcomes for the weather conditions.
2nd box
1st box
R
R
W
Rainy
B
RW
(
)
W
B
BR
(b) find the probability of getting
(i) 2 red balls,
Fine
Cold
(
)
(
)
(
)
Warm
(
)
(
)
(
)
Hot
(b) Find the probability that
(i) it is rainy and hot,
(ii) 1 red ball and 1 white ball,
Possible outcomes
Rainy and Cold
(
)
(
)
(
)
Cloudy and Warm
(
)
(
)
(
)
Fine and Hot
F.3 Mathematics Supplementary Notes
Chapter 12 Introduction to Probability
5/2006 P. 5
(ii) it is fine,
(iii) the numbers are equal,
(iii) it is warm or hot
(iv) the first number drawn is greater
than the second number
13.
12. Two numbers are drawn one by one from the
numbers 1, 2, 3 and 4 without replacement
(抽取後不放回).
(a) Complete the following tree diagram
to list all possible outcomes.
1st draw
1
2nd draw
2
3
4
Possible outcomes
1, 2
The figure shows a circular plate with 2
concentric circles. The radius of the outer
circle is twice that of the inner one. If a
fly stops on the plate at random, find the
probability that it stays on the shaded region.
(b) Find the probability that
(i) the sum of the two numbers is even,
14. A 3-digit number is in the form 34□, where
□ represents any integer from 0 to 9. Find
the probability of each of the following.
(a) the 3-digit number is odd,
(ii) the product of the two numbers is
even,
F.3 Mathematics Supplementary Notes
Chapter 12 Introduction to Probability
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(b) the 3-digit number is divisible by 3.
Answer
15. A school sells 200 tickets at $15 each. The
first prize is a $500 cash coupon and there are 10
prizes of $50 cash coupons. If I buy a ticket, find
the expected return.
16. An investment consultant tells Mr. Lee that
there is a 65% chance of making $20 000 profit on
a certain investment plan; otherwise, it is a total
loss. What is his expectation of profit?
1. (a) 12
(b)(i)
2. (a) 1
(b)
5. 21
6.(a)
1
10
(ii)
7
15
3.
5
6
(b)
2
3
1
3
3
10
7. (a)
3
5
(b)
1
2
8. (a)
1
52
(b)
1
13
(c)
1
26
9. (a)
1
3
(b)
1
4
(c)
1
2
1
9
11. (b)(i)
1
9
(ii)
1
3
(iii)
12. (b)(i)
1
3
(ii)
5
6
(iii) 0
14. (a)
1
2
(b)
13.
3
4
16. $13000
4.
3
4
(c)15
10. (b)(i)
(ii)
1
4
(iii)
2
9
(iii)
5
9
(iv)
1
3
(v)
2
3
3
10
(iv)
1
2
15. $5
4
9
F.3 Mathematics Supplementary Notes
Chapter 12 Introduction to Probability
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************************************OPTIONAL***************************************
Level III
1. A bag contains 3 red balls and 2 blue balls：R1 , R2 , R3 , B1 , B2
Two balls are drawn randomly one after the other without replacement.
(a) Find the probability that the two balls drawn are red.
(b) Find the probability that the first ball drawn is red and the 2nd ball drawn is blue.
(c) Find the probability that the first ball drawn is blue and the 2nd ball drawn is red.
(d) Find the probability that the two balls drawn are in different colors.
(e) Find the probability that no red ball is drawn.
(f) Find the probability that at least one blue ball is drawn.
(Ans.(a)
(f)
2.
3
3
3
3
1
(b)
(c)
(d) (e)
10
10
10
5
10
7
)
10
Peter has three $20 notes, two $50 notes and one $100 note in his wallet. If he draws two notes from
his wallet at random, one after the other and without replacement, find the probability that
(a) the notes drawn are of the same value;
(b) one of the notes drawn is a $20 note and the other is a $50 note
(c) none of the notes drawn is a $20 note
(d) the total value of the notes drawn is less than $120
(Ans.: (a)
3.
4
2
(b)
15
5
(c)
1
5
(d)
2
)
3
Three persons A, B and C each shoots at a target. The probabilities that the target is hit by A, B and C
are
1
1 1
,
and
respectively.
2 3
4
(a) Find the probability that
(i) all three persons hit the target
(ii) exact one person hits the target
(b) Among the three persons, one is chosen at random to shoot at the target. Find the probability
that he hits the target.
(Ans.: (a)(i)
1
11
13
(ii)
(b) )
24
24
36
F.3 Mathematics Supplementary Notes
Chapter 12 Introduction to Probability
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CE97-14
4. In a small pond, there were exactly 40 small fish and 10 large fish. The ranges of their weights W g
are shown in the following table:
Weight (W g)
Small fish
0 < W < 100
Large fish
500  W  600
In the morning on a certain day, a man went fishing in the pond. He caught two fish and their total
weight was T g. Suppose each fish was equally likely to be caught.
(a) Find the probability that
(i)
0 < T  200,
(ii) 500  T  700,
(iii) 1000  T  1200,
(iv) T > 1200.
(b) Suppose the two fish he caught in the morning were returned alive in the pond. He went fishing
again in the pond in the afternoon and also caught two fish.
(i) If the total weight of the fish caught in the morning was 650 g, find the probability that the
difference between the total weights of the fish caught in the morning and in the afternoon
is more than 200 g.
(ii) Find the probability that the difference between the total weights of the fish caught in the
morning and in the afternoon is more than 200 g.
(Ans.(a)(i)
156
245
(ii)
16
49
(iii)
9
245
(iv)0
(b)(i)
33
49
(ii)
29208
)
60025
F.3 Mathematics Supplementary Notes
Chapter 12 Introduction to Probability
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數學課外閱讀 :
【概率萬花筒】
蕭文強 林建 合著
出版社 : 廣角鏡
內容簡介 :
自然界的現象，概括地可分為必然現象和偶然現象。平日我們也一
定說過「這件事很可能發生」，「這不大可能吧」，粗略地說，概率論
是一門研究「可能性」的數學。這本小書透過大量有趣的概率的例子，
由淺入深，簡潔精要地介紹給讀者，比一般教科書常用的例子豐富得
多，這些例子不但擴闊讀者的數學視野，也使讀者活躍地思考。
F.3 Mathematics Supplementary Notes
Chapter 12 Introduction to Probability
5/2006 P. 10