This document provides an overview of probability and how it relates to statistics. It defines probability as a method for quantifying the likelihood of outcomes. Probability is measured as a ratio of the number of desired outcomes to the total number of possible outcomes. For outcomes to have a known probability, they must be selected through a random process. The normal distribution is discussed as it relates to probability, with common probabilities and areas under the normal curve defined. The document shows how to calculate probabilities for raw scores on a normal distribution using z-scores. It also demonstrates finding probabilities for ranges of scores and finding z-scores from known proportions.

1. PROBABILITY
Behavioral Statistics
Summer 2017
Dr. Germano

2. What is Probability?
• A method for measuring and quantifying the likelihood of
obtaining a specific sample from a specific population
I want to pick a single marble (a sample)
out of each of these jars (two populations)
What are my chances of getting a black marble from both jars?
Jar A Jar B50 white
50 black
10 white
90 black
You have a 50-50
chance of getting
either color
Your marble (your
sample) will
probably be black

3. What is Probability?
• A method for measuring and quantifying the likelihood of
obtaining a specific sample from a specific population
Now you are blindfolded when you pick your marble (your sample)
so you do not know from which jar (population) it comes from.
If you picked 4 black marbles in a row, which jar would say they came
from?
Jar A Jar B
Very high
probability they
came from this
one
Very low
probability they
came from this
one

4. Probability Defined
• The probability (p) of a specific outcome for any given
event is defined as the fraction or proportion of all the
possible outcomes (A, B, C, D, etc.)
p(A)= # of outcomes classified as A
total # of possible outcomes
A ratio that represents the likelihood that
some event will occur relative to the total
number of possible outcomes

5. Probability Defined
• If an event can occur in ‘A’ ways
• If an event can fail to occur in ‘B’ ways
• If all possible ways are equally likely (we guarantee this
through random selection)
• The probability of the event occurring:
• The probability of the event failing to occur:
 BA
A

 BA
B

p(A)= # of outcomes classified as A
total # of possible outcomes

6. A Demonstration
Select one card from a deck of 52 cards
What is the probability of selecting a king?
• The probability of the event occurring is computed by:
• A = 4
• there are 4 kings in the deck
• B = 48
• there are 48 cards that aren’t kings
in the deck
 BA
A
 52
4
)484(
4




7. A Demonstration
Select one card from a deck of 52 cards
What is the probability of selecting a king of diamonds?
The probability of the event occurring is computed by:
• A = 1
• There is 1 king of diamonds in
the deck
• B = 51
• There are 51 cards that are NOT
the king of diamonds in the deck
 BA
A
 52
1
)511(
1




8. Another way to think of probability
• Sample space: A set of possible outcomes
• Rolling a die: {1, 2, 3, 4, 5, 6}  6 possibilities
• Event: A subset of the sample space, the
occurrence of the thing you are interested in
• Die coming up odd: {1, 3, 5}  3 possibilities
Let’s say we are interested in the probability
a roll of a die will result in an odd number
The probability a die will come up odd is the number of event
possibilities over the number of sample space possibilities:
2
1
6
3
.#
.#
)( 
spacesample
iespossibilitevent
oddp

9. Interpreting Probability
• For any event (A), we assign a number p(A) between 0
and 1 that is interpreted as:
• The chance or likelihood that event A occurs
• The proportion of times that event A occurs over a series of trials
Continuing our rolling the die example…
p(odd) = ½ = 0.50 = 50%
We can express probability in a ratio, proportion, or %

10. Examples
If we spin this wheel: • What is the p of landing on 7?
• p(7) = 1/8
• p = 0.125 or 12.5%
• What is the p of landing on an
even number?
• p(even) = 4/8 = 1/2
• p = 0.50 or 50%
• What is the p of landing on a
white space?
• p(white) = 4/8 = 1/2
• p = 0.50 or 50%

11. More examples!
What is the probability of:
• Getting a correct answer on a multiple choice question
with five answer choices (assuming you make a blind
guess)?
• 1/5
• Getting a tail in
a regular coin toss?
• 1/2
• Rolling a 3 with a
regular die?
• 1/6

12. When is probability reliable?
• When you have a fair die, coin, roulette wheel, etc.
• There is an equal chance of every outcome
• If your coin is weighted to land on heads every time, the p(tail) is no
longer ½
To use probability in statistics, we must ensure that our
sample is chosen in a way to mirror this “fair chance”

13. Random Sampling
To insure that the definition of probability is accurate, we
must use random sampling
1. Every individual in the population of interest has an
equal chance of being selected
1. If more than one individual is to be selected for the
sample, there must be a constant probability for each
and every selection
• This may require sampling with replacement

14. An Example of Random Sampling
Imagine a large bag filled with coins.
The coins include half-dollars, quarters,
pennies, nickels, and dimes. You are
interested in estimating the weight of
this bag, but may not actually weigh every coin.
How would you estimate the weight?

15. Estimating the weight…
• Take a random sample of coins
• Every coin must have an equal chance
of being selected
• There must be a constant probability
for each and every selection
• Requires sampling with replacement
• Use this information to generalize what we might
expect the weight of the entire bag to be

16. The Role of Probability in Inferential Statistics
• Probability is used to predict what kind of samples are
likely to be obtained from a population
• Establishes a connection between samples and populations
• Inferential statistics rely on this connection when they use
sample data as the basis for making conclusions about
populations

17. Probability and Statistics
This is how we use probability in statistics
We think ‘backwards’ because we typically do not have the
population information
Based on the sample,
what do we estimate the
population to look like?
vs.
Based on the population’s
characteristics, what do we
estimate our sample to look like?

18. Probability and Frequency Distributions
• X: 1, 1, 2, 3, 3, 4, 4, 4, 5, 6
• N = 10
If I sample n = 1:
• p(X < 2)?
• 2/10 = 1/5
• p(X > 3)?
• 5/10 = 1/2
• p(X = 3)?
• 2/10 = 1/5

19. PROBABILITY AND THE
NORMAL DISTRIBUTION

20. 68.26%
94.46%
99.73%
Probability and the Normal Distribution
We can describe a normal distribution by the
proportions of area contained in each section
• Symmetric: Proportions on the right side will correspond
to the mirror image on the left
• Since we can identify sections with
z-scores, these proportions
correspond to any normal
distribution (regardless of
actual mean and standard
deviation values)

21. μ
34.13%
13.59%
2.14%
Because the z-distribution is normally distributed we
can make certain assumptions about probabilities
The z-score allows us to
draw a vertical line
through the distribution,
dividing the distribution
into two sections
The Body The Tail

22. μ
34.13%
13.59%
2.14%
Some probabilities associated with the
normal distribution
• 34.13% of the distribution falls between the mean and the
first standard deviation (z = +1.00; z = -1.00)
• 13.59% of the distribution falls between the 1st and 2nd
standard deviation
• 2.14% of the distribution falls
between the 2nd and 3rd
standard deviations

23. Probability and
the Normal
Distribution
It is good to have a
general idea of
proportions within the
curve
What if you need to find
the proportion below a z-
score of -0.85?
The Unit Normal Table
(Appendix B, pp.699-
702) lists z-scores and
their corresponding
proportions
Note: The Unit Normal Table lists positive z-scores only because the curve
is symmetric; values can be used for negative z-scores as well.

24. The Unit Normal Table
• A negative z-score: tail is on the left side
• A positive z-score: tail is on the right side
Because the Unit Normal Table only lists positive numbers, it is helpful to draw
the curve and the area you want before looking up the proportion so you know
which value (body or tail) you want

25. Finding the probability of a z-score
What is the proportion of the normal distribution associated
with the following sections of a graph:
• z > 1.25
Interpreted as:
• The probability of selecting a participant from this population with a
z-score greater than 1.25 is 0.1056
• The proportion of the distribution with a z-score greater than 1.25 is
10.56%
• z < 0.00
Interpreted as:
• The probability of selecting a participant from this population with a
z-score less than 0.00 is 0.5000
• The proportion of the distribution with a z-score less than 0 is 50%
p = 0.1056
p = 0.5000

26. PROBABILITIES AND
PROPORTIONS FOR SCORES
from a Normal Distribution

27. Finding the probability of a raw score from
a normal distribution
What is the probability of randomly selecting a
person with an IQ greater than 115 if the
population μ=100 and σ=10?
1st step: Calculate the z-score
2nd step: Find the proportion of the curve greater
than the z-score.
• z = 1.50
• p(>1.50) = p in the tail (Column C) for 1.50 = 0.0668
50.1
10
)100115(


z

28. You Try It:
For a distribution with a μ = 100 and σ = 10:
What is the probability of randomly selecting a
person with an IQ less than 90? Greater than 125?
00.1
10
)10090(


z 50.2
10
)100125(


z
Proportion in the tail above z = 2.50:
p = 0.0062
This very high z-score indicates it is at the
extreme high end of the distribution, with few
scores above
Proportion in the tail below z = -1.00:
p = 0.1587

29. Finding the Probability for a Range of
Scores
What is the probability of randomly selecting a person
with an IQ between 95 and 110? (μ=100 σ=10)
• Find the two z-scores, and then find the proportion that
covers the space between these two scores
00.1
10
)100110(
50.0
10
)10095(






z
z
Add the distances of each z-score from
the mean (Column D):
(0.1915 + 0.3413) = 0.533
Or, subtract the proportion in the tail of
one score from the proportion in the
body of the other score (e.g., Body of
110 – Tail of 95)
(0.8413 - 0.3085) = 0.533
p = 95< IQ <110( )

30. Finding the z-score
corresponding to a certain proportion
• If you know a specific location (z-score) in a normal
distribution, you can use the table to look up
corresponding proportions.
• If you know a specific proportion(s), you can use the table
to look up the exact z-score location in the distribution

31. Example (1)
• What z-score separates the lowest 5% from the
remainder of the distribution?
• Find the z-score with p = 0.05 in the tail
• z-table shows us z = 1.65 has p = 0.0495
(closest to p = 0.05 without exceeding it)
• Remember we want the lowest 5%, and the table only
shows us positive z-scores
• Since the curve is symmetric: a z-score of z = -1.65
separates the lowest 5% from the remainder of the
distribution

32. Example (2)
• What z-scores separate the extreme 5% scores
from the rest of the distribution?
• We want the lowest and highest in this 5%
• Lowest score separates (5%/2) = 2.5% lowest and
highest score separates (5%/2) = 2.5% highest
• We want to find the z-scores with p = 0.025 in tails
• z-scores of z = +/- 1.96 separate these scores from the
rest of the distribution

33. Solving for a specific score:
• If I want the top 10%, what is the cutoff score?
(μ = 100 σ = 15)
• z-score that separates p = .0100 closest without
going over is z = 1.29
• Now I want to solve for X to find the cutoff score
2.1191002.19100)15(28.1  zX

34. Recall from earlier, we can express cumulative
percentages as percentile ranks.
To find the 84th percentile we would find the score that separates p = 0.8400
below (body) and p = 0.1600 above (tail). (approximately z = 0.99)

35. Looking ahead to inferential statistics
• Based on our observations of a sample, we make
inferences about the population.
• It is with statistics based on probability that we determine
how likely our sample represents our population of
interest
• or, how unlikely our sample is compared to the population after
some treatment/intervention/etc.