Download Probability

Probability
I’ve often wondered what were the chances of
Dorothy’s house falling directly on the wicked Witch
of the East?
Why Probability?
• The problem facing statisticians is to infer what
characteristics would remain if random sampling
variability were eliminated.
• To answer this, we need to know what kind of sample
results can be expected on the basis of chance alone
• How about an example from ESP?
Please think of one of these six cards and concentrate on it.
Please think of one of these six cards and concentrate on it.
Ps yc hology: An Introduc tion
Charles A. Morris & Albert A. Mais to
© 2005 Prentic e Hall
Now, write down the name of your card on a piece of
paper, but do not show it to anyone just yet.
Continue to concentrate, but do not “force” your
thoughts… just relax and hold the image of the card in
your mind.
I will now remove the card which I believe is yours from
the set of
six.
I will
now remove the card from the set of six which I
believe is yours.
Please read what card you have written down on the
paper…
yourcard
card you
shouldhave
not written
be amongdown
these. on
Please read
what
paper… your card should not be among these.
Ps yc hology: An Introduc tion
Charles A. Morris & Albert A. Mais to
© 2005 Prentic e Hall
the
Uncertainty
• Probability gives us a way of expressing uncertainty that
an event will occur
• Probabilities are typically presented in several ways
– percents (e.g., there a 20% chance of rain today)
– proportions (e.g., the prob. of Tom getting a hit is .29)
– fractions (e.g., the prob. of getting a “head” is 1/2).
Approaches to Probability
• There are three approaches to probability
– classical (or logical)
– empirical relative-frequency
– subjective
• Each approach has advantages and disadvantages and
supplements the others
Classical Approach
• The probability of an event is given by the number of
outcomes of interest divided by the total number of
possible outcomes under consideration
What is the probability of rolling a “3” on a single die?
Given there is one outcome of interest (i.e., a “3”) and
a total of six possible outcomes under consideration
(i.e., a “1,” “2,” “3,” “4,” “5,” and “6”)
P(6) = 1/6 = .17
Assumptions of Classical Approach
• There are two assumptions made when calculating
probabilities with the classical approach
– each outcome is equally likely
– probabilities are based on an infinite series of trials
• Those assumptions, however, are often not met in real life
A Counting Problem
• One difficulty encountered is knowing how many possible
outcomes there are to consider
– For example, given four basketball teams at the local
youth center, what’s the probability the end-of-season
standings will be
1st place 2nd place 3rd place 4th place -
Bill’s Barbers
Mike’s Meats
George’s Garage
Lou’s Laundry
• Permutations and combinations help answer such questions
Permutations
• When you have a set of values (e.g., teams) and need to
know in how many different ways they can be ordered, you
are asking about permutations (order is important)
nPn
= n!
The symbol “!” represents factorial: n(n-1)(n-2)…1
In this example,
4 x 3 x 2 x 1 = 24
Permutations Continued
• When you have a set of values and need to know in how
many different ways some of them can be ordered, you are
asking about the permutation of n objects taken r at a time
n!
nPr =
(n-r)!
For example, how many different 1st and 2nd place
finishes could there be?
In this example,
4x3x2x1
4!
=
= 12
4P2 =
2x1
(4-2)!
Combinations
• When you have a set of values and need to know in how
many different ways some of them can be grouped
together, you are asking about the combination of n objects
taken r at a time (order is not important)
n!
C
=
n r
r!(n-r)!
For example, given eight students, how many ways
could they be put together as lab partners?
In this example,
4
8x7x6x5x4x3x2x1
8!
=
= 28
8C2 =
2 x 1 (6 x 5 x 4 x 3 x 2 x 1)
2! (8-2)!
Empirical Relative-Frequency Approach
• The probability of an event is given by the number of times
the outcome of interest occurs divided by the number of
sampling experiments conducted
What is the probability of rolling a “3” on a single die?
Outcome
1
2
3
4
5
6
Frequency
3
5
5
3
4
4
n = 24
P(3) = 5/24 = .21
Drawbacks to Empirical Approach
• The true probability is approached as the number of
sampling experiments approaches infinity
• Data is not always available
Subjective Approach
• The probability of an event is based on one’s expectations
that the event will occur
• Expectations may arise from personal experience,
intuition, expertise, etc
• May be only means of assigning a probability in some
situations
Problems with Subjective Approach
• Different people are likely to assign different probabilities
to the same event
Basic Concepts
• Simple event (Ei) - a single outcome (e.g., rolling a “3”)
• Compound event - an event that can be broken down into
two or more simple events (e.g., “an even number”)
• Sample space - set of all sample points
– Assign a sample point to each simple event
Sample Space for Rolling a Die
E1
E3
E2
E5
E4
E6
Euler Diagram
• Euler diagrams can be used to illustrate the calculation of
probabilities
Sample Space for Rolling a Die
E1
P(number < 4)
= 3/6 = .5
E3
E2
E5
E4
E6
P(even number or number <4)
= 5/6 = .83
P(even number)
= 3/6 = .5
More Concepts
• Union of events ( ) - the probability of event A or event B
or both A and B
P(A)
P(A)
Intersection
of A and B
Addition Rule: P(A B) = P(A) + P(B) - P(A B)
An Example
What is the probability of rolling an even number (event A) or
a number less than five (event B) or both?
Sample Space for Tossing a Coin
event B
E1
both A
and B
E3
E2
E5
E4
E6
P(A B) = 3/6 + 4/6 - 2/6 = 5/6
event A
• When events are mutually exclusive -- one outcome
precludes another outcome from occurring -- the individual
probabilities are added
– For example, the probability of rolling a number less
than 3 (event A) or greater than 4 (event B)
Addition Rule for
Mutually Exclusive events: P(A B) = P(A) + P(B)
= 2/6 + 2/6 = 4/6
Even More Concepts
• Intersection of events or joint occurrence ( ) - the
probability of event A and event B
– For example, we might want to know the probability of
• getting three consecutive “heads”
• being a psychology major and an internship
participant
• having a heart attack and having no family history of
heart disease
• The calculation of probabilities for the joint occurrence of
events depends on whether the events are independent or
not.
Independent Events
• When events are independent -- the outcome of one event
has no influence on the outcome of another event -- the
individual probabilities are multiplied
Multiplication Rule
for Independent Events: P(A B) = P(A) P(B)
For example, the probability of rolling three consecutive
“4s”
P(A B
C) = P(A) P(B) P(C) = 1/6 x 1/6 x 1/6 = 1/216
• There are some instances when the outcomes are not
independent and the outcome of one event does influence
the outcome of another event
A
B
Intersection
of A and B
Multiplication Rule
for Non-independent Events: P(A B) = P(A) P(B|A)
= P(B) P(A|B)
Conditional Probabilities
• Conditional probability (A|B or B|A) -- the probability
of one event occurring given that another other event has
occurred.
– For example, selecting a card from a deck influences
the probability of the next card
P(A B)
P(A|B) = P(B)
P(A B)
P(B|A) = P(A)
Illustration of Conditional Probability
Lowerclassmen (B)
n = 50
Females (A)
n = 30
100 Students
P(A B)
P(A|B) =
P(B)