Subject: MDM4UI
Topic: Theoretical Probability (2)
Unit: 1.4
Homework: Page 218
6, 7, 8, 10, 12, 14
Objective:
The study of probability began with the analysis of the games of chance and then the
understanding of the mathematics was advanced to other areas
Definition:
Sample space:
the collection of all possible outcomes of an experiment
Example: An experiment consists of rolling a single die and observing the up face
The sample space S is S = {1, 2, 3, 4, 5, 6}
Event Space:
the collection of all outcomes of an experiment that correspond to a
particular event. of interest.
Example: An experiment consists of rolling a single die and observing the up face when
it has an even number.
The event space A is A = {2, 4, 6}
Simple Event:
An event that consists of exactly one outcome
Example: There is only one way of rolling a 4 on a dice
Theoretical
Probability:
the ratio of the number of outcomes that make up that event to the
total number of possible outcomes
P ( A) =
n (A)
n ( S)
where S is the sample space and A is the event space
and n(A) are the number of element in the event space and n(S) are
the number of elements in the sample space.
Probability of a Complementary event: If A is an event in a sample space, the probability of
the complementary event A′ , is given by
P ( A′ ) = 1 − P ( A ) .
Example: Many board games involve a roll of two six-sided dice to see how far you may move
your pieces or counters. What is the probability of rolling a total of 7?
To calculate the probability of a particular total, count the number of times it appears
in the table. For event A {rolling a 7).
Die
Totals
1
2
3
4
5
6
1
2
3
4
5
6
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
n ( A)
n(S )
n ( rolls totalling 7 )
=
n ( all possible rolls )
6
=
36
1
=
6
P ( A) =
The probability of rolling a total of 7 is
1
.
6
Example: A messy drawer contains 3 red socks, 5 white socks, and 4 black socks. What is the
probability of not drawing a red sock?
It is easier of find the probability of drawing a red sock and then finding its
complementary (not drawing a red sock).
n ( A)
n(S )
n ( drawing a red sock )
=
n ( drawing any sock )
3
=
12
1
=
4
P ( A) =
Thus, P ( A′ ) = 1 − P ( A )
1
= 1−
4
3
=
4
There is a
3
or a 75% chance that you will not pick out a red sock.
4