Download Chapter 7 Sets & Probability

Chapter 7
Sets & Probability
Section 7.3
Introduction to Probability
Why use probability?
A great many problems that come up in
the applications of mathematics involve
random phenomena – those for which
exact prediction is impossible.
The best we can do is determine the
probability of the possible outcomes.
Sample Spaces
In probability, an experiment is an activity
or occurrence with an observable result.
Each repetition of an experiment is called
a trial.
The possible results of each trial are called
outcomes.
The set of all possible outcomes for an
experiment is the sample space for that
experiment.
Example: A spinner that is equally divided up
into three spaces is spun and a coin is tossed.
Give the sample space for this experiment.
Spinner
Coin Toss
H
Possible Outcomes
(1, H)
1
T
(1, T)
H
(2, H)
T
(2, T)
H
(3, H)
T
(3, T)
2
3
S = { (1,H), (1,T), (2,H), (2,T), (3,H), (3,T) }
Events
An event is a subset of a sample space.
An event in which only one outcome is
possible is called a simple event.
If the event equals the sample space, then the
event is called a certain event.
If the event is equal to the null, or empty, set,
then the event is called an impossible event.
Example: Consider rolling a single die.
S = { 1, 2, 3, 4, 5, 6}
Event A: Rolling a 5
Event B: Rolling an odd number
Event C: Rolling a number less than 7
Event D: Rolling a number greater than 6
Which event is a simple event?
Event A
Which event is an impossible event?
Event D
Which event is a certain event?
Event C
Set Operations for Events
Let E and F be events for a sample space, S.
E  F occurs when both E and F occur;
E  F occurs when E or F or both occur;
E ′ occurs when E does not occur.
Mutually Exclusive Events
Events E and F are mutually exclusive events
if E  F = Ø
Probability
For sample spaces with equally likely outcomes, the
probability of an event is defined as follows.
Basic Probability Principle:
Let S be a sample space of equally likely
outcomes , and let event E be a subset of S. Then
the probability that event E occurs is
For any event E, 0 ≤ P(E) ≤ 1.
Example: A marble is drawn from a bowl
containing 3 yellow, 4 white, and 8 blue
marbles. Find the probability of the following
events.
1.) A yellow marble is drawn
P (yellow) = 3 / 15 = 1 / 5
2.) A blue marble is drawn
P (blue) = 8 / 15
3.) A white marble is not drawn
P (not white) = 11 / 15