CHAPTER 8
LESSON 1
Teachers Guide
What is a Distribution?
Counting (as opposed to measuring)
Objective:
 To introduce the concept of a (discrete) probability distribution.
Investigate
For each of the following experiments compare the following two distributions:
 the frequency distribution obtained by performing the experiment repeatedly.
 the probability distribution obtained from the theoretical probabilities.
1. Roll a die
Probability Distribution
for One Roll of a Die
6
Probability
Frequency
Typical Frequency Distribution
for 24 Rolls of a Die
4
2
0
1
2
3
4
5
Numbers on the die
6
1/6
0
1
6
1
6
1
1
6
3
1
6
1
6
1
6
5
Numbers on the die
111111
666666
Observations:
 When you roll a die 24 times do you always get the same frequency distribution as the one
shown above? Explain. No – You would expect some variation on the frequency counts
due to chance fluctuations.

When you roll a die 24 times is it possible that you get a frequency distribution that has the
same “flat top” shape as the probability distribution? Yes – However this is unlikely.
If this happened, what would be the height of each of each bar on the frequency scale?
1
(24) = 4
6

What is the total area of all the bars in the probability distribution?
Total area = 1

In this example the probability distribution is called a uniform distribution because;
Each outcome has the same probabilities.
Probability Distribution
3 Flips of a Coin
Probability
2. Flip a coin 3 times
Frequency
Typical Frequency Distribution
3 Flips of a Coin - 16 Repetitions
0
3/8
1/4
1/8
0
8.0
2
Number of Heads in 3 Flips
4.0
1
8
0.0
0
1
2
3
3
8
3
8
1
8
Number of Heads in 3 Flips
Observations:
 What is the total area of all the bars in the probability distribution? Total area = 1

Which of the following probability methods could be used to calculate the probabilities
shown in the graph of the probability distribution? Choose one and calculate the
probabilities.
Sample Space
1
HHH
8
Probability Tree
1
2
HHT
HTH
3
8
H
THH
1
2
HTT
THT
3
8
H
P (HHH) =
TTH
T
1
8
T
1
2
H
T
1
2
1
2
1
2
1
2
H
x
 1  1
P(x) = 3Cx    
2 2
1
2
H
1
2
1
2
Binomial Formula
T H
3
8
3
2T) =
8
1
2
T
1
2
1
2
T H
1
2
T
P (TTT) =
P ( 2H) =
P(
3- x
1
8
1
8
* 8 equally likely outcomes
TTT
Probability Distribution (Discrete)
Definition
A probability distribution is - A function that gives the probability for each outcome of the
experiment
Different ways to express a Probability Distribution 1) Table 2) Graph 3)Function (formula)
Kinds of Discrete Probability Distributions
1. Uniform Distribution
A discrete uniform probability distribution is a probability distribution that gives equal
probability to each outcome.
Example: One spin of the pointer
$5
Express the probability distribution of the value the pointer comes up at as
$10 $15
a) a table
b) a graph (histogram)
$0
c) a function
P(x) =
1
4
1
4
0
5
10 15
( note: width of bars is 1)
For x = 0,5,10,15
2. Binomial Distribution
For an experiment with
 a fixed number of independent and identical trials
 only two possible outcomes (“success” or “failure”) per trial.
the binomial distribution gives the probability of getting x “successes” in n trials is
P(x) = nCx pxq(n-x)
where p = the probability of a “success” on a single trial
* calculator =binompdf (n,p,x) q = the probability of a “failure” on a single trial
Example: Spin the pointer 4 times
$5
Express the probability distribution of the number of times
that the $10 comes up as
$0
$10 $15
a) a table
b) a graph (histogram)
c) a function
1 3
P( x)  4Cx    
4 4
x
Value
.4
Probability
.3
0
81 / 256 ~ .32
1
108 / 256 ~ .42
2
3
4
 3  = 81

 4  256
4
P(O) = 
.2
3
108
 1  3 
  =
256
 4  4 
.1
54 / 256 ~ .21
12 / 256 ~ .05
1 / 256 ~ .004
4
P(1) = 4 
0
1
2
2
3 4
2
54
1 3
P(2) = 6     =
256
4 4
1
X =(-0.5,4.5) Y=(-0.15,0.6)
L1 ,0,1,2,3,4
L2 = binompdf (4,
1
, L1)
4
3
 1   3  12
  =
 4   4  256
P(3) = 4 
4
1
1
P(4) =   =
256
4
3. General Discrete Probability Distribution (No Name!)
The uniform and binomial are two important probability distributions that have been given
names because they have special probability patterns. However many applications produce
probability distributions that don’t have names and are not uniform or binomial.
Example: Spin the pointer 2 times
$5
Express the probability distribution of the sum of the 2 spins.
$10 $15
a) a table
Values
b) a graph (histogram)
c) here the function
representation is
not convenient
Probability
0
5
1 / 16
2 / 16
10
3 / 16
15
4 / 16
20
3 / 16
25
2 / 16
30
1 / 16
0
5 10
$0
15 20
25 20
Width= 1
Need width = 1…..so P(x) =1
2nd spin
Sample space
1st spin
0
5
10
15
0
0,0
0,5
0,10
0,15
5
0,5
5,5
5,10
5,15
10 10, 0 10,5
10,10
10, 15
15 15,0
15,10
15,15
15,5
16 equally likely outcomes
Summary
For any discrete probability distribution

0  P ( x)  1
for every value of x

 P (x )  1
(the sum of all the probabilities is 1)