This document discusses probability and simple events. It defines key probability concepts like simple events, sample spaces, and events. It provides an example of rolling a fair six-sided die and identifies the simple events, sample space, and examples of non-simple events. It then covers computing probabilities of events using formulas, properties of probabilities, mutually exclusive and independent events, and the complement of an event. Formulas for addition and multiplication rules are provided along with examples.

1. Math 409/409G
History of Mathematics
The Probability of
Simple Events

2. What is probability?
Probability is the measure of the
likelihood of a random phenomenon or
chance behavior.

3. Definitions
A simple event is any single outcome
from a probability experiment.
A sample space, S, of a probability
experiment is the collection of all simple
events.
An event is any collection of outcomes
from a probability experiment. An event
consist of one or more simple events.

4. Example
A probability experiment consists of rolling
a single “fair” die.
• What are the simple events of this
probability experiment?
• What is the sample space?
• Give two examples of events that are not
simple events.

5. Answers
• The simple events are the possible
outcomes of rolling the die. Since there
are 6 numbers on the die, the simple
events are:
 “Rolling a 1”  {1}
 “Rolling a 2”  {2}
 “Rolling a 3”  {3}
 “Rolling a 4”  {4}
 “Rolling a 5”  {5}
 “Rolling a 6”  {6}

6. • The sample space, S, is the set of all
simple events. So
 S  {1, 2, 3, 4, 5, 6}
• An event consists of one or more simple
events. So two non-simple events are:
 E  {2, 4, 6}  “Rolling an even number”
 O  {1, 3, 5}  “Rolling an odd number”

7. Another Definition
• A probability experiment is said to have
equally likely outcomes if each simple
event in the sample space has the same
chance (probability) of occurring.
Example: When rolling a “fair” die, each
number on the die has the same chance of
occurring as any other number. But if you
replace the number 6 on the die with the
number 2, then you no longer have equally
likely outcomes since rolling a 2 has a
better chance or occurring.

8. Computing Probabilities
If an experiment has n equally likely
simple events and if the number of ways
that event E can occur is m, then the
probability of event E is
number of ways that can occur
Pr [ ]
total number of possible outcomes
E
E
m
n



9. Examples
A probability experiment consists of rolling
a single “fair” die. Then:
1
6
Pr [rolling a 2] 
3 1
6 2
Pr [rolling an even number]  
4 1
6 3
Pr [rolling a no. larger than 2]  
0
1 0
6
Pr [rolling a ]
  

10. Important Properties
If S is the sample space and E is an event
in that sample space, then:
1
Pr [ ]
S 
0 1
Pr [ ]
E
 

11. Mutually Exclusive Events
• Two events are mutually exclusive if they
cannot occur at the same time.
Example: When rolling a single fair die,
the events of rolling a 1 and of rolling a 2
are mutually exclusive events since the die
will show only one number. But the events
of rolling a 1 and of rolling an odd number
are not mutually exclusive since 1 is an
odd number.

12. Addition Rules
• If A and B are mutually exclusive events,
then
• If A and B are not mutually exclusive
events, then
Pr[ or ] Pr[ ] Pr[ ]
A B A B
 
Pr[ or ] Pr[ ] Pr[ ] Pr[ and ]
A B A B A B
  

13. Examples
• When rolling a single fair die, what’s the
probability of rolling a 1 or a 2?
1 1 1
0
6 6 3
Pr[1 or 2] Pr[1] Pr[2] Pr[1 and 2]
  
   

14. • When rolling a fair die, what is the probability
of rolling a 1 or rolling an odd number?
First note that event of rolling a 1 and rolling
an odd number is the same as the event of
rolling a 1.
1 3 1 1
6 6 6 2
Pr[1 or odd] Pr[1] Pr[odd] Pr[1 and odd]
  
   

15. Independent Events
• Two events A and B are independent if
knowing whether A occurs does not
change the probability that B occurs.
Example: Two marbles are drawn one at
a time from an urn containing 3 red
marbles and 2 blue marbles. Are the
events of first drawing a red marble and
then drawing another red marble
independent events?

16. Well, that depends on whether or not the
first marble is placed back in the urn.
• If the marble is put back in the urn after
drawing the first marble, then the event of
first drawing a red marble and then
drawing another red marble are
independent events since after putting the
marble back, the sample space remains
the same. So the probability of drawing the
second red marble is the same as the
probability of drawing the first red marble.

17. • But if the first marble is not put back, then
after the first marble is drawn, the sample
space has been reduced by one marble.
So the probability of the second marble
has changed after drawing the first marble.
So the events of first drawing a red marble
and then drawing another red marble are
not independent events.

18. Multiplication Rules
• If A and B are independent events, then
• If A and B are not independent events,
where
Pr[ and ] Pr[ ] Pr[ ]
A B A B
 
|
Pr[ and ] Pr[ ] Pr[ ]
A B A B A
 
|
Pr[ ] probability of given
that has occured.
B A B
A


19. Examples
• Two marbles are drawn from an urn
containing 3 red marbles and 2 blue
marbles. What’s the probability of drawing
two red marbles if the first drawn marble is
placed back in the urn?
|
3 3 9
5 5 25
Pr[ and ] Pr[ ] Pr[ ]
R R R R R
 
  

20. • Two marbles are drawn from an urn
containing 3 red marbles and 2 blue
marbles. What’s the probability of drawing
two red marbles if the first drawn marble is
not placed back in the urn?
|
3 2 6 3
5 4 20 10
Pr[ and ] Pr[ ] Pr[ ]
R R R R R
 
   

21. The Complement of an Event
The complement, Ec, of event E is the
event that E does not occur.
Example: If E is the event of rolling an
even number on a fair die, then Ec is the
event of rolling an even number.

22. Complement Rule
• For any event A, the probability that A does
not occur is
Sometimes it is easier to use the
complement of an event than it is to use the
actual event. In this situation you want to use
1 ].
Pr[ ] Pr[
c
A A
 
1 ].
Pr[ ] Pr[ c
A A
 

23. Example
• If you randomly select a number between
(and including) 1 and 100, what is the
probability that that number is less than 99?
If E is the event in this problem, it would be
easier to look at Ec, the event of selecting 99
or 100. So in this case,
2 98 49
1 ] 1 .
100 100 50
Pr[ ] Pr[ c
E E
     

24. This ends the lesson on
The Probability of
Simple Events