Transcript Introducing Probability

INTRODUCING
PROBABILITY
A probability model has two components:
A sample space and an assignment of probabilities.
This is denoted with an S and is a set
whose elements are all the possibilities
that can occur
Each element
of S is called
an outcome.
A probability of an outcome is a number and
has two properties:
1. The probability assigned to each outcome is
nonnegative.
2. The sum of all the probabilities equals 1.
Let's roll a die once.
This is the sample space---all the possible outcomes
S = {1, 2, 3, 4, 5, 6}
probability an
event will occur
Number of ways that E can occur
PE 
Number of possibilities
What is the probability you will roll an even number?
There are 3 ways to get an even number, rolling a 2, 4 or 6
3 1
P  Even number   
6 2
There are 6 different numbers on the die.
The word and in probability means the intersection of two events.
What is the probability that you roll an even number and
a number greater than 3?
E = rolling an even number
2 1
PE  F   
6 3
F = rolling a number greater than 3
How can E occur?
{2, 4, 6}
How can F occur?
{4, 5, 6}
E  F  {2, 4, 6}  {4,5, 6}  {4, 6}
The word or in probability means the union of two events.
What is the probability that you roll an even number or a
number greater than 3?
4 2
PE  F   
6 3
E  F  {2, 4, 6}  {4,5, 6}  {2, 4,5, 6}
ADDITION RULE
For any two events E and F,
P(EF) = P(E) + P(F) - P(EF)
Let's look at a Venn Diagram to see why this is true:
If we count E
EE
FF
and then count F,
we've counted the things in both twice
so we subtract off the intersection
(things in both).
ADDITION RULE for Mutually
Exclusive Events
If E and F are mutually exclusive events,
P(EF) = P(E) + P(F)
Mutually exclusive means the events are disjoint.
This means E  F = 
Let's look at a Venn Diagram to see why this is true:
E
F
You can see that since
there are not outcomes
in common, we won't be
counting anything twice.
E
This is read "E complement" and
is the set of all elements in the
sample space that are not in E
Remembering our second property of probability,
"The sum of all the probabilities equals 1" we can
determine that:
PE  PE 1
This is more often used in the form
P  E   1 P  E 
If we know the probability of rain is 20% or 0.2
then the probability of the complement (no rain)
is 1 - 0.2 = 0.8 or 80%
Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au