Download Probability

Probability
The Study of
Chance!

When we
think about
probability,
most of us
turn our
thoughts to
games of
chance
Probability is also the
underlying foundation for the
methods we use in
inferential statistics.
Let’s learn some
basics
PROBABILITY
BASICS
Any action with unpredictable
EXPERIMENT:
outcomes
SAMPLE SPACE:
The set of all possible outcomes
“S”
EVENT:
A specific outcome
P (A):
The probability of outcome A
occurring
Complement:
The complement of any event “A”,
consists of all outcomes in which
“A” does NOT occur.
PROBABILITY
BASICS
•A probability is a number between 0 and 1
For any event A, 0 ≤ P(A) ≤ 1
•The probability of the set of all possible outcomes
(sample space) must be 1.
P(S)=∑P(A) = 1
This is known as
“The something’s gotta happen rule”
•The probability of an impossible event is 0
•Most events have probabilities
somewhere between!
•The probability of an event that is certain is 1
Approaches to Probability

Relative Frequency Approximation:
(Experimental Probability)
 Conduct the experiment a large number
of times and record the results
Then:
number of times A occurred
P( A) 
number of trials
For Example
??What’s the probability that I roll a 4
with a standard die (A) ??
To answer this
using relative
frequency
approach
(experimental
probability) I
rolled a die
200 times.
Outcomes
Outcome
Frequency
1
2
3
4
5
6
30 37 30 29 34
40
# times "4" 29
P(4) 

 .145
# rolls
200
nd
2

Approach to Probability
Classical Approach to Probability
(Theoretical Approach)
**Requires Equally-likely Outcomes**
 Assume that a procedure has “n”
different simple events and each of those
events has an equal chance of happening
Then:
number of ways A can occurr
P( A) 
number of possible events
For Example
??What’s the probability that I roll a 4
with a standard die (A) ??
Let’s answer
this using
the relative
frequency
approach
(theoretical
probability)
•A standard die has one “4”
# of 4' s
1
P(4) 
  .166
# of possibilit ies 6
•A standard die has 6 possibilities
So, let’s summarize
Experimental
Theoretical
P(4) = .145
P(4) = .166
Why did we get two different answers?
Probability is really about what would
happen “in the long run”. The “Law of
Large Numbers” tells us that experimental
probability will approach the actual
(theoretical) probability in a LARGE
number of trials. If we increased the
number of rolls our probability would get
closer and closer to .166
rd
3

Approach to Probability
Subjective Probability
 Probabilities are found by simply
guessing or estimating its value based on
prior knowledge of the relevant
circumstances
•Examples of subjective probability
include forecasting the weather
Complementary Events

The complement of an event is when the
event does NOT happen.
Remember that
one of our basics
of probability is
the “something’s
gotta happen rule”
which says that
the sum of all
possibilities equals
1.
So, the probability of a
complement (A’) can be found
with 1 – the probability that the
event happened.
P(A’) = 1 – P(A)
For Example
For a single card drawn from a standard deck of
cards, what is the probability that it is NOT a
diamond?
We can use what we know about
Facts about
complementary events to answer
cards:
this question.
13 diamonds
13 hearts
13 spades
13 clubs
Call “diamond” event A
P(A’) = 1 – P(A)
P(A’) = 1- (13/52)= .75
Multiple Events

Consider the event:
• Tossing a coin followed by rolling a
standard die
• We want to calculate the probability of
rolling a 5.
• To find this probability, consider the
following model.
We will use a tree diagram to represent the stages in the event. In
order to keep the “tree” manageable, we will define each stage as
simply as we can. i.e.—”success” or “not a success”
The second stage was to roll a die.
Although there are 6 possible outcomes,
we are only interested in the outcome
“roll a 5”. We can simplify our tree by
designating only “success”—5, and not a
success—5c
The first
stage has two
possible
outcomes
1
6
Remember: Our
question is
P(rolling a 5) = ?
5
P(heads & 5) =
5c
Notice this happens in two
places in our tree.
 1  1 
  
 2  6 
Heads
5
6
1
2
Toss a
coin
1
2
1
6
P(tails & 5) =
5
Tails
5
6
 1  1 
  
 2  6 
5c
To find the probabilities for these
two “branches”, we multiply the
individual probabilities across and
then add the two branches together
Calculating Probability

This means that the probability of
rolling a “5” is calculated by:
• = (1/2)(1/6) + (1/2)(1/6)
• = (1/12) + (1/12)
• = (2/12)
Multiple Events and a couple of
shortcuts

Let’s consider the following situation
• Roll a standard die 3 times

Now find the following probabilities
• What is the probability that we get one “4”
in 3 rolls?
• What is the probability that not all rolls
are “4’s”
• What is the probability that we get at
least one “4”?
Draw a Picture
Find the
probability that
we get one “4”
in three rolls.
1
6
4
1
6
5
6
4c
4
1
6
4
1
6
5
6
4c
4c
5
6
Roll a
Die
5
6
4
5
6
4c
5
6
2nd Roll
1
6
4c
Find the
probabilities for
each branch
(1/6)(5/6)(5/6)=.1157
4
1
6
4
1
6
Find the
branches where
this happens.
5
6
3rd Roll
4c
(5/6)(1/6)(5/6)=.1157
4
(5/6)(5/6)(1/6)=.1157
.3471
4c
Then find the sum
of the
probabilities of
these branches
Draw a Picture
Find the
probability that
not ALL are
“4’s”
1
6
4
1
6
5
6
4c
4
1
6
4
1
6
5
6
4c
4c
5
6
Roll a
Die
5
6
4
4
1
6
4
1
6
5
6
4c
4c
5
6
2nd Roll
1
6
4c
5
6
3rd Roll
4
4c
1-[(1/6)(1/6)(1/6)]=.9954
Find the
branches where
this happens
Find the
probabilities for
each branch
OR notice that all
branches except
the branch where
all are 4’s are
checked…..
So….we could
find the P(not all)
by using
complements
1- P(all)
Draw a Picture
Find the
probability that
at least one is a
“4”
1
6
4
1
6
5
6
4c
4
1
6
4
1
6
5
6
4c
4c
5
6
Roll a
Die
5
6
4
4
1
6
4
1
6
5
6
4c
4c
5
6
2nd Roll
1
6
4c
5
6
3rd Roll
4
4c
Find the
branches where
this happens
Find the
probabilities for
each branch
OR notice that all
branches except
the branch where
none are 4’s are
checked…..
So….we could
find the P(at
least one) by
using
complements
1- P(none)
1-[(5/6)(5/6)(5/6)]=.4213
Additional Resources

The Practice of Statistics—YMM
• Pg 310 -322

The Practice of Statistics—YMS
• Pg 328-355

Against All Odds—Video #15
• http://www.learner.org/resources/series65.
html