Sta220
Section 3.1
__________________ is a measure of the likelihood of a random phenomenon or chance
behavior. Probability describes the long-term proportion with which a certain _____________
will occur in situations with short-term uncertainty.
The Law of Large Numbers
As the number of repetitions of a probability experiment increases, the proportion with which a
certain outcome is observed gets closer to the probability of the outcome.
An _____________________ is an act or process of observation that leads to a single outcome
that cannot be predicted with certainty.
A _____________________________) is the most basic outcome of an experiment.
Example: Listing Sample Point for a Coin-Tossing Experiment
Two coins are tossed, and their up faces are recorded. List all sample points for this experiment.
The ___________________, S, of a probability experiment is the collection of all possible
outcomes.
An _______________________ is any collection of outcomes from a probability experiment. An
event may consist of one outcome or more than one outcome. We will denote events with one
outcome, sometimes called simple events, ei. In general, events are denoted using capital letters
such as E.
Example: Identifying Events and the Sample Space of a Probability Experiment
Consider the probability experiment of having two children.
(a) Identify the outcomes of the probability experiment.
(b) Determine the sample space.
(c) Define the event E = “have one boy”.
The probability of sample point is a number between ________________ which measures the
likelihood that the outcome will occur when the experiment is performed.
Probability Notation:
• Probabilities can be expressed as a ___________ (always reduced), ___________, or
_______________.
•
P(A) = 0.123 means the probability of event A is 0.123
•
Unless noted, round probabilities to _________ decimal places.
A __________________________________ lists the possible outcomes of a probability
experiment and each outcome’s probability. A probability model must satisfy rules 1 and 2 of the
rules of probabilities.
Example: A Probability Model
In a bag of peanut M&M milk chocolate candies, the colors of the candies can be brown, yellow,
red, blue, orange, or green. Suppose that a candy is randomly selected from a bag. The table
shows each color and the probability of drawing that color. Verify this is a probability model.
Color
Probability
Brown
0.12
Yellow
0.15
Red
0.12
Blue
0.23
Orange
0.23
Green
0.15
If an event is ________________, the probability of the event is 0.
If an event is a _______________, the probability of the event is 1.
An ______________________ is an event that has a low probability of occurring.
EXAMPLE Building a Probability Model
Pass the PigsTM is a Milton-Bradley game in which pigs are used as dice. Points are earned based
on the way the pig lands. There are six possible outcomes when one pig is tossed. A class of 52
students rolled pigs 3,939 times. The number of times each outcome occurred is recorded in the
table at right.
(Source: http://www.members.tripod.com/~passpigs/prob.html)
Outcome
Frequency
(a) Use the results of the experiment to build a probability
model for the way the pig lands.
(b) Estimate the probability that a thrown pig lands on the
“side with dot”.
(c) Would it be unusual to throw a “Leaning Jowler”?
Side with no dot
1344
Side with dot
1294
Razorback
767
Trotter
365
Snouter
137
Leaning Jowler
32
Example: Die Tossing Experiment
A fair die is tossed, and the up face is observed. If the face is even, you win $1. Otherwise, you
lose $1. What is the probability that you win?
Example: Probability of an Event-Coin-Tossing Experiment
Consider the experiment of tossing two unbalanced coins. Because the coins are not balanced,
their outcomes ( H or T) are not equiprobable. Suppose the correct probabilities associated with
the sample points are given in the accompanying table. [Note: The necessary for assigning the
probabilities to sample points are satisfied.] Consider the event
A: {Observe exactly one head}
B: {Observe at least one head}
Calculate the probability of A and the probability of B
Sample Point
Probability
HH
4/9
HT
2/9
TH
2/9
TT
1/9
Probability of an Event
The probability of an event A is calculated by _______________ the probabilities of the sample
points in the sample space for A.
Steps for Calculating Probabilities of Events
1. ___________________________: that is, describe the process used to make an
observation and the type of observation that will be recorded.
2. ___________ of the sample points.
3. Assign ______________________ to the sample points.
4. ________________ the collection of sample points contained in the even of interest.
5. ________________ the sample point probabilities to get the probability of the event.
Example: Selecting 5 Movies from 20
Suppose a movie reviewer for a newspaper reviews 5 movies each month. This month, the
reviewer has 20 new movies from which to make the selection. How many different samples of 5
movies can be selected from the 20?