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Introduction to Behavioral
Statistics
Probability, The Binomial
Distribution and the Normal Curve
Probability, The Binomial Distribution and
the Normal Curve

Introduction to Probability
– We all think in terms of probability
– We all compute and use probability
– If we have a coin, there is a 1/2 probability that it
will land on heads/tails when we flip it. (.5 heads +
.5 tails gives coin a total probability of 1)
– What is the probability of drawing a particular card
from a deck. (1/52 or .019 or 1.9 per 100)
– What is the probability of drawing any two (2)
cards from a deck. (1/52 + 1/52=2/52 or .038)
Probability, The Binomial Distribution and
the Normal Curve

Additive Theorem – The probability that any one of a set of
mutually exclusive events will occur is the
sum of the probability of the separate
events.
– What is the probability of drawing either of
two cards from a deck.
– (1/52 + 1/52) = 2/52 or .038)
Probability, The Binomial Distribution and
the Normal Curve

Multiplication Theorem – The joint probability of obtaining both of two events is the product of
their separate probabilities.
 What is the probability of drawing two aces from a deck of cards.
– (4/52 * 3/51) = 12/2652 or .0045)


What is the probability of first a 5 and then a 6.
– (1/6 * 1/6) = 1/36
With two dice, what is the probability of rolling a 7 or 11?
– How many ways to get 7 (4 +3) (5+2) (6+1) on each die
» using the additive theorem we see that there are
6/36 ways to get 7 and 2/36 to get 11
» Thus there are 1/9 ways to to roll a 7 or 11.
Probability, The Binomial Distribution and
the Normal Curve

Multiplication Theorem 
What is the probability of drawing two aces from a deck of cards.
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Probability, The Binomial Distribution and
the Normal Curve

Permutations of r objects taken r at a
time
N
  N ! 9! = 362880
N

Permutations of N objects taken r at a
time
N
N!
N!



 72
 N  r  !  9  2 !
r
Probability, The Binomial Distribution and
the Normal Curve

Combination of N objects taken r at a
time.
N
N!
10!
3628800
C 


 45
r r! N  r! 2!10  2! 240320
Probability, The Binomial Distribution and
the Normal Curve

Binomial Distribution
– This distribution is very important to
psychology.
The chi square distribution is based on it…
 The normal curve is based on it….
 F and t distributions are The based on it...

Probability, The Binomial Distribution and
the Normal Curve

Binomial Distribution
 Bernoulli Trial

Experiments often have only two possible
outcomes.
– true false
– effective not effective

Flipping a coin one time and noting whether it
lands heads or tails, or randomly drawing one
sample from a distribution is called a Bernoulli
trial or Bernoulli experiment.
Probability, The Binomial Distribution and
the Normal Curve

Bernoulli Trial
– Characteristics of a Bernoulli trial



A trial can result in one of two outcomes
The probability of success remains constant from trial to
trial.
The outcomes of successive trials are independent.
– In reality very few real situations meet these
requirements since probability doesn’t remain
constant when we remove an item from the
distribution.
Probability, The Binomial Distribution and
the Normal Curve

Binomial Distribution
– In this distribution, the random variable is a
sum (the number of successes observed
on n greater than or equal to two Bernoulli
trials.

This distribution is a relatively simple example
of an important class of theoretical distributions
or models that are referred to as sampling
distributions.
Probability, The Binomial Distribution and
the Normal Curve

Binomial Distribution
– Sampling distribution is the special name
that is given to a probability distribution
where the random variable is a statistic
based on the result of n greater than or
equal to 2 trials.
– The Binomial Distribution is one of the
distributions used by psychologists.
Probability, The Binomial Distribution and
the Normal Curve

Binomial Distribution
– The number of successes observed on n greater than or
equal to 2 identical Bernoulli trials is called a binomial
random variable, and its probability distribution is called a
binomial distribution.
– If we toss a fair coin 5 times, the probability of observing
exactly r heads in n tosses is given by p(X=r)= nCrprqn-r
– This gives the probability that the random variable X equals r
heads. nCr is the combination of n objects taken r at a time.
P is the probability of a success (a head), and q = (1-p).
– Next - lets look at a particular example:
Probability, The Binomial Distribution and
the Normal Curve
1   1 

p ( X  4 ) C

 

 2   2 
 1   1   5
5 !

 

4 ! 5  4  !  2   2 
3 2
4
5
5  4
4
4

1
Binomial Distribution for N=5 and p=1/2
No. of Heads
p(X=r)
0
1
2
3
4
5
1/32 5/32 10/32 10/32 5/32 1/32
Probability, The Binomial Distribution and
the Normal Curve
Binomial Distribution for N=5 and p=1/2
No. of Heads
p(X=r)
0
1
2
3
4
5
1/32 5/32 10/32 10/32 5/32 1/32
Histogram for binomial distribution shown above….
Notice how much this
resembles the form of a
normal curve.
Probability, The Binomial Distribution and
the Normal Curve


The normal curve is a
limiting form of the
binomial distribution
The normal curve
occurs when we have
an infinite number of
events occurring
according to the laws of
chance.
10
9
8
7
6
5
4
3
2
1
0
5
4
3
2
1
0
Probability, The Binomial Distribution and
the Normal Curve
A Linear Graph
y=A+bX

60
50
– In this case we have
used the formula
y=A+bX to plot a
straight line.
40
30
20
10
0
We can write a
formula to plot a set
of points

In this same way we
can generate a plot
of the normal curve
Probability, The Binomial Distribution and
the Normal Curve
1
f  X 
 2
X    

e

2
/ 2
Where:
f(X)= height of the distribution at X
Pi = approximately 3.142
e=
base of natural logarithms)
approximately 2.718
2
Probability, The Binomial Distribution and
the Normal Curve
Fortunately, we don’t need to calculate the normal
curve.
We just use the table in the back of our book…..
(1)
Standard
Score
(2)
Area from
Mean to
Standard
Score
(3)
Area in Larger
Portion
(4)
Area in
Smaller
Portion
(5)
y ordinate at X
0.00
.0000
.5000
.5000
.3989
1.00
.3413
.8413
.1587
.2420
1.96
.4750
.9750
.0250
.0584
Using the Normal Curve to normalize a
distribution of scores
(1 )
S ta n d a rd
S c o re
z
x

(2 )
A re a fro m
M e a n to
S ta n d a rd
S c o re
(3 )
A r e a in L a r g e r
P o r tio n
(4 )
A r e a in
S m a lle r
P o r tio n
(5 )
y o r d in a te a t X
0 .0 0
.0 0 0 0
.5 0 0 0
.5 0 0 0
.3 9 8 9
1 .0 0
.3 4 1 3
.8 4 1 3
.1 5 8 7
.2 4 2 0
1 .9 6
.4 7 5 0
.9 7 5 0
.0 2 5 0
.0 5 8 4
so z  x or z   X  X 
so
z  X  X
Using the Normal Curve to normalize a
distribution of scores
(1 )
S ta n d a rd
S c o re
(2 )
A re a fro m
M e a n to
S ta n d a rd
S c o re
(3 )
A r e a in L a r g e r
P o r t io n
(4 )
A r e a in
S m a lle r
P o r t io n
(5 )
y o r d in a t e a t X
0 .0 0
.0 0 0 0
.5 0 0 0
.5 0 0 0
.3 9 8 9
1 .0 0
.3 4 1 3
.8 4 1 3
.1 5 8 7
.2 4 2 0
1 .9 6
.4 7 5 0
.9 7 5 0
.0 2 5 0
.0 5 8 4
z  X  X
1. We find 90th centile from column 3 of table.
2. We then use the corresponding z score.
3. In case of 90th centile z=1.29
4. Using IQ data 1.29(20.2)+103.29= 129.348
This is the normalized centile score.
Determining the % of cases which fall
between any two scores
 For our IQ data - suppose we want to know what %
of scores fall between 123.49 and 113.32.
 First we convert these scores to z scores
(123.49-103.29)/20.2 = 1.00
(113.32-103.29)/20.2= .50
 Then we get the area from column 2 and subtract
.3413-.1915=.1498 or 15%
 To see exactly where the above values came from,
use the table in your book and work through this
problem.
Using the table to determine the expected
frequency of any given score


Lets suppose a shirt maker wants to determine how
many shirts of a given neck size should be made.
We will assume:
– Average neck size is 15 and the SD is 2.
– formula:

Fe=(iN/F)y
– Where
» I=size of interval
» N=number of shirts

fe=(1000/2).3532 =176 size 16 shirts.
The Normal Curve and Z scores: Some
Final Considerations



The term normal curve implies that this
type of curve is normal.
– Mathematicians did once believe that this curve
was ‘normal’ and that is how it got its name.
– We now know that this is a ‘chance’ distribution,
not a normal distribution.
The normal curve extends from ± infinity as the line
never actually touches the base line
±1 sd locates the deflection point for the normal
curve. (line moves out more than down)
Well-that's it. Next we will look at
correlation.
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Introduction to Behavioral
Statistics
Probability, The Binomial
Distribution and the Normal Curve