This document provides an overview of key concepts in probability. It defines probability as a measure of likelihood or chance that can be estimated using relative frequency or subjective estimates. It introduces common probability notation and outlines three basic rules for computing probability: relative frequency approximation, classical approach, and subjective probabilities. Examples are provided to illustrate these concepts and rules. The document also discusses the law of large numbers, probability limits between 0 and 1, and guidelines for rounding off probabilities.

1. Essentials of
STATISTICS
Chapter 3 Probability

2. A measure of likelihood
or chance
Can be estimated by checking
relative frequency
or how often an
occurrence can be
expected?
What is probability?

3. There is a 30% chance of
rain tonight.
1 in every 2 marriages ends in
divorce
Slightly more than half of all
births result in males.
Second hand smoke increases
a person’s chance of
developing lung cancer by 15%.
44% of people believe in
intelligent life on other
planets

4. Where does probability
fit in …
…with statistics?
What’s the deal with probability?

5. Statisticians are looking for
answers….
What’s the deal with probability?
….explanations for things that
happen ….

6. Suppose a possible explanation
for an observed characteristic in
data is isolated….
What’s the deal with probability?
….a large number of New Jersey children
suffering from leukemia…

7. What’s the deal with probability?
…all live within 3 miles of
a nuclear power plant ….

8. What’s the deal with probability?
….Then the statistician can determine
through probability the likelihood that
the explanation is, in fact, the “cause” of
what was observed….

9. Rare Event Rule
If, under a certain assumption, the probability of an
observed event is extremely small, then the
assumption is probably NOT correct.

10. Accept explanations
associated with HIGH
probability
Reject explanations
associated with LOW
probability
Rare Event Rule

11. Let’s assume that the lottery is a fair game of chance.
Then, suppose you won the Wisconsin state lottery five
times in a row. What would happen?
That’s right, you’d be rich! (Really, really rich!) But,
everyone around you would think the lottery was rigged,
and you were cheating!
Why?
Because of the Rare Event Rule!
Example of the Rare Event Rule

12. Rare Event Rule
If, under a given assumption, the probability of a
particular event is really, really SMALL,
but we perform an experiment and observe that
rare event occurring, then as statisticians we
would conclude that the assumption is NOT
correct.

13. The “statistician” hired by the doubting lottery losers
would conclude that…
Because the likelihood of you winning the lottery five
times in a row is virtually zero.
0.00000000000000000000000000000000000000000001
(I made that number up! But it’s really, really close to zero, isn’t it?)
The assumption of a “fair” lottery must be rejected.
In other words, you cheated!
Impact of Rare Event Rule on Lottery Example…

14. Event - the result(s) or outcome(s) from
some procedure or experiment
Simple event - any outcome or event
that cannot be broken down into
simpler components
Sample space - all possible simple
events
We need some definitions…

15. (1) Consider the experiment of having a child.
The simple events are girl or boy.
The sample space is { girl, boy}
Some Examples:
(2) What happens when you roll a die?
The simple events are 1, 2, 3, 4, 5, or 6 dots.
The sample space is { 1, 2, 3, 4, 5, 6}.

16. Notation
P – Uppercase “P” denotes a probability
A, B, ... – Other uppercase letters represent
specific events
P (A) - represents the probability of event A

17. Example Using New Notation
Suppose you are interested in the
probability of having a girl.
Then the simple events become
A = girl is born
B = boy is born

18. Example Using New Notation
You would write the “probability of a
female birth” as P(A) and the
“probability of a male birth as P(B).

19. Example Using New Notation
Suppose you are interested in the
likelihood that college students cheat
on exams.
Then the simple events become
A = student cheated on exam
B = student did not cheat on exam

20. Example Using New Notation
How would you write
“probability of a student cheating”?
Answer:
P(A)

21. Time to put some numbers…
….On all of those
“fancy” ideas…..

22. Basic Rules for
Computing Probability
Rule 1: Relative Frequency Approximation
Conduct (or observe) an experiment a large number of
times, and count the number of times event A actually
occurs, then an estimate of P(A) is
P(A) = number of times A occurred
number of times trial was repeated

23. Relative Frequency Approximation
Example:
What is the likelihood of rolling a die
and getting a “6”?
The “relative frequency” response
would be to ….

24. Example:
Roll a die a zillion times (okay, let’s just do it
50 times!) and count the number of “6”s that
show up…
Why don’t YOU go do that….I’ll wait here!
Relative Frequency Approximation

25. Suppose you rolled 50 times and found
10 sixes showed up.
P(A) =
number of times 6 occurred
number of times die was tossed
10
50
= = 0.20 or 20%
Relative Frequency Approximation

26. On the basis of our die experiment, we would
expect a six to “pop up” about 20% of the
time.
So, if you rolled a die 400 times, you would
expect to get a six about
400 x 20% = 80 times.
Relative Frequency Approximation

27. Basic Rules for
Computing Probability
Rule 2: Classical approach
Start with a procedure that has n different
possible outcomes, each with an equal
chance of occurring.
Let s be the number of ways event A can occur
Then, the classical approach would tell you that
the probability of A occurring is found by ….

28. Basic Rules for
Computing Probability
Rule 2: Classical approach
P(A) =
number of ways A can occur
number of different simple
events
s
n
=

29. Classical Approach
Let’s visit our rolling die again…
The classical approach to probability would say
there are 6 equally likely outcomes when you roll
a die.
{ 1, 2, 3, 4, 5, 6 }
And, in those six outcomes, there is 1 chance of
getting a six.

30. Classical Approach
The probability of rolling a die and getting a
six is found by …
P(six)=
1
6
= .16666 ≈.167 or 16.7%

31. Basic Rules for
Computing Probability
Rule 3: Subjective Probabilities
P(A), the probability of A, is found by simply
guessing or estimating its value based on
knowledge of the relevant circumstances.
This might also be known as the “educated guess”!

32. Subjective Probability
One example of subjective
probability that each of us
deal with every day on the
evening news is the
weather report….
There is a 30% chance of rain
showers on Tuesday morning.

33. The relative frequency approach is an
approximation based on experimentation.
Relative Frequency vs Classical vs Subjective
The classical approach is exact.
Subjective is an educated guess.

34. Law of Large Numbers
As a procedure is repeated again and
again, the relative frequency probability
(from Rule 1) of an event tends to
approach the actual probability.

35. Sample space = { 0,1,2,3,4,5,6 }
These simple events are equally likely.
 you can use classical probability to discover
that the probability of a six is 1/6. However, let’s
experiment with the Law of Large Numbers…
Roll the die 10 times, then 100 times, then 1000 times,
then 10,000 times…..see what happens…..
Law of Large Numbers: An Example
What is the probability of rolling a die and getting a “six”?

36. Law of Large Numbers: An Example
What is the probability of rolling a die and getting a “six”?
Number Rolls Number Sixes
(simulated)
Relative
Frequency
10 3 3/10 = 0.3
100 22 22/100 = 0.22
1000 160 160/1000 =
0.160
10,000 1625 1625/10,000 =
0.1625

37. Law of Large Numbers: An Example
What is the probability of rolling a die and getting a “six”?
Notice as the number of observations increases,
the relative frequency probability gets closer and
closer to the “true” probability, or classical
probability, of 1/6 = 0.16667.
0.30  0.22  0.160  0.1625

38. Sample space = { struck, not struck }
These simple events are not equally likely.
 you cannot use classical probability
You must use the relative frequency
approximation (Rule 1) or subjectively
estimate the probability (Rule 3).
Law of Large Numbers: An Example
What is the probability that an individual is struck by lightening?

39. Find an “Answer” Using the Relative Frequency
Approach to Probability….
We can research past events to determine that in
a recent year 377 people were struck by
lightning in the US, which has a population of
about 274,037,295.
Law of Large Numbers: An Example
What is the probability that an individual is struck by lightening?

40. P(struck by lightning in a year) ≈
377 / 274,037,295 ≈ 1/727,000
Law of Large Numbers: An Example
What is the probability that an individual is struck by lightening in the
United States?
This probability will be pretty close to the “true” probability of begin
struck by lightening, because it is based on the observations of a
really large number of people.

41. Probability Limits
 The probability of an impossible event is
0.
P(Thanksgiving is on Wednesday this year) = 0
 The probability of an event that is certain
to occur is 1.
P(Thanksgiving is on Thursday this year) = 1
Be cautious as you calculate probabilities
and remember the “limits”!

42. Probability Limits
 The probability of an impossible event is 0.
 The probability of an event that is certain
to occur is 1.
0 ≤ (A) ≤ 1

43. Possible Values for Probabilities
1
0.5
0
Certain
50-50 Chance
Impossible
Likely
Unlikely

44. Rounding Off Probabilities
 When you calculate a probability, give the
exact fraction or decimal
1/2 2/3 0.125
or
 …if you choose to show the decimal
approximation, round off the final result
to three significant digits
2/3 = 0.66666….. ≈ 0.667