# Unit 7A

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```					          Unit 7A

Fundamentals of Probability
Outcomes & Events
Example: Over a period of 8 years, a family has 3
children. Each birth is an event. List the different
possible outcomes that would result in the event of
2 boys and 1 girl.

• Outcomes are the most basic possible results of
observations or experiments.

• An Event consists of one or more outcomes that
share a property of interest………so an Event is a
specific type of outcome.
Counting Outcomes
The Multiplication         Principle is a method for counting outcomes.
For a sequence of two events in which the first event can occur M ways and the second
event can occur N ways, the events together can occur a total of M × N ways.

Example 1: How many outcomes are possible when you flip a coin twice?
Example 2: How many outcomes are possible when you roll two dice?

The Multiplication Principle holds for more than two events.
Example 1: How many outcomes are possible when you flip a coin 3 times?
Example 2: The sports car you want come with or without a sunroof, with or
without power windows, and in red, black, or green. How many different
versions of the car are available?

Try #23 – 30
Probability
• Every event has a certain probability, or likelihood
of happening.

• The probability of an event is expressed as
P (event).

• Usually upper case letters are used to represent
events.
For example, we could say that A is the event of
getting “heads” when a coin is tossed.
So, P(A) would be the probability of getting
“heads” when a coin is tossed.
Probability Limits
• The probability of an
impossible event is 0.

• The probability of an event that
is certain to occur is 1.

• Any event A has a probability
between 0 and 1, inclusive.

0 ≤ P(A) ≤ 1
Three Types Of Probabilities

I. Theoretical Probabilities

II. Empirical Probabilities

III. Subjective Probabilities
I. Theoretical Probability
A theoretical probability is based on a model in
which all outcomes are equally likely.
Let A be an event.
number of ways A can occur
P( A) 
total number of outcomes

Example: What is the probability of having 2 boys and 1
girl when having 3 kids?

Try #31 - 47
Example
Find the probability of rolling a “7” when a pair of fair dice are tossed.

number of ways a 7 can be rolled
P(rolling7)                                   
total number of outcomes
                                                                       
                                                                                                
                                                                           

                                                                                         
                                                                                              
                                                                                                 

                                                                                                 
                                                                                                
                                                                                         

                                                                                           
                                                                                              
                                                                                           

                                                                                           
                                                                                                   
                                                                                           

                                                                                           
                                                                                              
                                                                                           

Do you remember how to count outcomes?
II. Empirical Probability
An empirical probability is based on observations
or experiments. It is the relative frequency of the
event of interest.
Examples of Empirical Probabilities include:
Baseball batting averages.               hits
BA 
total at bats

FT % 
free throws attempted
Computing An Empirical Probability

Conduct (or observe) a procedure a large number
of times, and count the number of times that event
A actually occurs. Based on these actual results
P(A) is estimated as follows:

number of times A occurred
P( A) 
number of times A could have occurred
Example
An allegedly fair die was tossed 563 times. The
number “4” occurred 96 times. What is the
probability that the next toss will result in a „4‟?
If you toss a genuinely fair die, what is the
probability of tossing a “4”?

Try #49 – 54
III. Subjective Probability
A subjective probability is an estimate based on
experience or intuition.

Examples:
An economist was asked “What is the probability that the
economy will fall into recession next year?” The economist
said the probability was about 15%.

There‟s a 60% chance that UGA will win the national
championship this year.
Probability Of An Event Not Occurring
Suppose the probability of an event A is P(A).
P(A) + P(not A) = ______
What is the probability that the event A does not
occur?

P(not A) = 1 − P(A)
Try #57 – 64
Probability Distribution of an Event
A probability distribution represents the
probabilities of all possible events.

In this course, you will make probability
distributions in the form of a table.

One column of the table lists each event and the
other column lists each probability. The sum of
all the probabilities must be 1.
Making A Probability Distribution
Step 1: List all possible outcomes….outside the table.
Step 2: List each event in the 1st column of the table.
Step 3: Find the probability of each event using the
theoretical method.
Step 4: List the probabilities in the 2nd column of
the table. Remember that the sum of all the
probabilities must be 1.
Example: Suppose you throw a pair of dice. Let x be the
sum of the numbers on the dice. Make a table for the
x       P(x)
probability distribution of the
2   1/36 ≈ 0.028
event x.
3   2/36 ≈ 0.056
4   3/36 ≈ 0.083
5   4/36 ≈ 0.111
6   5/36 ≈ 0.139
7   6/36 ≈ 0.167
8   5/36 ≈ 0.139
9   4/36 ≈ 0.111
10   3/36 ≈ 0.083
11   2/36 ≈ 0.056
12   1/36 ≈ 0.028
Total 36/36 =1.000
Example: Suppose you toss a coin twice. Let x be the total number of
tails on the tosses. Make a table for the probability distribution of x.

Example: Suppose a coin is tossed thrice. Make a probability
distribution for the number of heads.
Number Probability

Total

Total

Try #65 – 66.
Odds
The odds for (or odds in favor of) an event A:

# chances for A : # chances against A

The odds against (or odds on) an event A:

# chances against A : # chances for A
Examples:
1. What is the probability of rolling a 4?
What is the probability of not rolling a 4?
The odds for rolling a 4 on a die is:
The odds against rolling a 4 on a die is:

2. What is the probability of drawing a 7 from a deck of cards?
What is the probability of not drawing a 7?
What are the odds for drawing a 7?
What are the odds against drawing a 7?

Try #67 – 70
Odds In Gambling

In gambling, the “odds on” (odds against) usually
expresses how much you can gain with a win for
each dollar you bet.

EXAMPLE: At a horse race the odds on the
horse Median are given as 5 to 2. If you bet \$12
on Median and he wins, how much will you gain?

Try #71 – 72

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