Download What is Probability

Name : ________________(
)
Class : 4M2
Date : __________
Self Study : Probability (Sec 4 IM)
Passport 1:
Click on this link to find our more on:
1) Properties on Probabilty:
http://www.stat.psu.edu/~resources/ClassNotes/hrm_06/sld005.htm
2) Probability of a single event:
http://www.cimt.plymouth.ac.uk/projects/mepres/book7/bk7i21/bk7_21i2.htm
3) Probability of combined events:
http://www.bbc.co.uk/schools/ks3bitesize/maths/handling_data/probability_2_2.shtml
4) Independent and Mutually Exclusive Event:
Independent : http://regentsprep.org/Regents/Math/mutual/Lindep.htm
Mutually Exclusive: http://regentsprep.org/Regents/Math/mutual/Lmutual.htm
Complement : http://regentsprep.org/Regents/Math/mutual/Lcompl.htm
5) Probability Tree :
http://www.youtube.com/watch?v=PVF5QBMF4lk
http://www.ict-teacher.com/Mathstutor/Probability.html#anchor622353
6) Set notation on Probability:
http://math.arizona.edu/~alexa/m263/setsym.pdf
Congratulation, you have completed this self study module.
Passport 2:
You must proceed to read and complete the practice questions in the notes attached. You
must use the discussion thread to gain enlightenment to your queries. I had posted 6
threads related to the practice questions 1 – 6. Kindly reply in the post all your doubts.
(Assessment = 10% collaboration online)
Passoprt 3:
To check your understanding on Probability, you have to construct a concept map
usingCmap 2.0. Click on this attachment to check on the rubics for concept map. Please
export Cmap as image file and completed concept map must be uploaded in IVLE
workbin students submission by 26th May 2009 by 5pm. Format to save your file:
ProbabilityCmapAngCC4M216.jpeg or ProbabilityCmapAngCC4M216.jpg.
(Assessment = 20% concept map)
Passport 4:
Complete the assignment and upload into IVLE by 27th May 2009 by 5pm.
(Assessment = 10% assignment)
Passport 5:
Take the short test on probability on 29th May 2009.
(Assessment = 60%)
Page 1 of 7
Name : ________________(
)
Class : 4M2
Date : __________
What is Probability?
Probability is the study of chance or the likelihood of an event happening. Directly or
indirectly, probability plays a role in all activities. In probability, any observation or
measurement of a random phenomenon is an experiment.
The possible results of the experiment are called outcomes and the collection of all possible
outcomes is called the sample space.
Definition:
An experiment is a process or investigation from which results are observed or recorded.
An outcome is a possible result of an experiment.
A sample space is the set of all possible outcomes in the experiment. It is usually denoted by the
letter S . Sample space can be written using the set notation, { }.
The probability of an event =
n( E )
number of outcomes of the event
i.e. P( E ) 
n( S )
total number of possible outcomes
Example 1
A die numbered from 2, 4, 6, 8, 10 and 12 is tossed. Find the probability of getting a 4.
Sample space = {2, 4, 6, 8, 10, 12} implies total number of possible outcomes is 6.
P ( getting 4) 
1
6
Example 2
A coin is tossed, find the probability that it will land tails up.
Sample space = {Head, Tail} implies total possible outcomes is 2. P (tails up ) 
1
2
Practice 1:
Box B contains 3 red balls, 2 yellow balls and 7 white balls.
A ball is drawn at random from box B. What is the probability that the ball drawn is
i) white?
Total outcome =
i) P(white) =
ii) red?
iii) yellow?
ii) P (red) =
iii) P(yellow) =
What are the Properties of Probability?
Let P ( E ) denotes the probability of an event E happening, then
Page 2 of 7
Name : ________________(
)
Class : 4M2
Date : __________
(a) 0  P ( E )  1
In other words it is impossible to get negative probability or probability greater than one.
(b) If E is an impossible event, then P( E )  0 .
(c) If E is a certain event then P ( E )  1
Example 3
A die numbered from 1 to 6 is tossed. Find the probability of getting a 9.
P( getting 9)  0
Example 4
A bag has 3 red marbles. A marble is randomly selected, what is the probability that a red ball is
selected.
P(red ball selected )  1
Practice 2:
Box A contains 4 red balls and 5 yellow balls. What is the probability of getting a black ball?
P(black) =
Three Rules of Probability
(I) Addition Rule of Probability (for mutually exclusive events)
Two events are called mutually exclusive if the occurrence of one precludes the other event from
occurring.
In a Venn diagram, the two sets would not intersect and the chance of them happening at the
same time is zero. In other words, P( A  B)  0 i.e. P( A  B)  P( A or B)  P( A)  P( B) .
Example 5
A die numbered from 1 to 6 is tossed. Find the probability of getting either 2, 5 or 6.
P(2  5  6)  P(2)  P(5)  P(6) 
1 1 1 1
  
6 6 6 2
Practice 3:
A cubical die suspected of being biased is thrown a large number of times. It is found that the
probabilities of obtaining 2, 3, 4 or 5 are equal and is found to be
obtaining 1 is found to be
Number
Probability
1
. The probability of
8
1
. Complete the following table showing the probabilities :16
1
2
1
16
3
1
8
Page 3 of 7
4
5
6
Name : ________________(
)
Class : 4M2
Date : __________
(II) Multiplication Rule of Probability (for independent events)
Two events are said to be independent if one event has no effect on the probability of the other.
If A and B are independent events, then P( A and B)  P( A  B)  P( A)  P( B)
Example 6
A bag contains two red marble and three white marble. A marble is chosen at random and then
put back into the bag. If the process is repeated twice, find the probability that
(i) a red marble was chosen every time,
2 2 4
P( Red  Red )  P( Red )  P( Red )       
.
 5   5  25
(ii) a red marble was chosen followed by a white ball.
 2 3 6
P( Red , then white)  P( Red )  P( white)       
 5   5  25
(iii) each colour was chosen.
P(each colour chosen)
= P(1st red, 2nd white) or P(1st white, 2nd red)
= P(1st red, 2nd white) + P(1st white, 2nd red)
 2 3
 5 5
12
=
25
=   
+
3  2
  
5  5
Practice 4:
A bag contains two red marble and three white marble. If two marbles are chosen from the bag,
find the probability of
(i) getting two red marbles,
(ii) getting one red and one white marbles.
(III) Complementary Rule
Page 4 of 7
Name : ________________(
)
Class : 4M2
Date : __________
If the probability that an event will occur is P ( E ) and the probability that it will not occur is
denoted by P( E ) then P( E )  1  P( E )
Example 7
A die numbered from 1 to 6 is tossed. Find the probability of not getting a 4.
P(not getting 4)  1  P( getting 4)  1 
1 5
 .
6 6
Note : This is a very useful rule to bear in mind when to compute P( E ) directly is very tedious.
Practice 5:
Each evening, Mr Tan either watches television or reads a book. The probability that he watches
television is
3
. What is the probability that he reads a book?
5
P(read a book) =
Some useful tools for Probability
(I) Probability Tree Diagram
In a tree diagram, the probability is written on the branches while the outcomes are written at the
ends of the branches.
A coin is tossed twice. The tree diagram below shows the possible outcomes and their
probabilities.
1st toss
1
2
H
1
2
T
2nd toss
1
2
H
1
2
T
1
2
H
1
2
T
Example 8
Page 5 of 7
Name : ________________(
)
Class : 4M2
Date : __________
A bag has 4 red marbles and 6 white marbles. Peter takes two marbles at random from the bag,
one after the other, and are not replaced.
(i)
Draw a tree diagram to show the possible outcomes and their probabilities.
(ii)
Find the probability that
(a) the two marbles selected are of the same colour,
(b) one marble will be red and the other white.
(i)
1st marble
4
10
6
10
(ii)
(a)
R
W
2nd marble
3
9
R
6
9
W
4
9
R
5
9
W
P(both same colour)
= P(both red) + P(both white)
=
 4  3   6  5  7
       
 10  9   10  9  15
 4  6   6  4  8
      
 10  9   10  9  15
(b) P(both different colour) = P(red, white) + P(white, red) = 
(II) Possibility Diagram
Example 9
A tetrahedral die A are marked 2, 4, 6 and 8 on its four faces.
A tetrahedral die B are marked 1, 3, 5 and 7 on its four faces.
The two dice are thrown together and their scores added. Find the probability that the total score
is a prime number.
Solution:
The question could be solved easily if we could present all the possible total score systematically
in the form of a possibility diagram.
+
1
3
5
7
2
3
5
7
9
4
5
7
9
11
6
7
9
11
13
8
9
11
13
15
From the possibility diagram, it is obvious that 11 out of 16 total scores are prime numbers.
Page 6 of 7
Name : ________________(
P(total is a prime) = 
)
Class : 4M2
Date : __________
11
16
Practice 6:
A box contains 12 cards of which 4 are red and 8 are white. Two cards are drawn randomly from
the box.
a)
Complete the probability tree below.
b)
Find the probability that both cards are red.
c)
Find the probability that the colour of the two cards is different.
a)
Red
4
12
Red
8
12
White
White
Red
White
Page 7 of 7