Download Chapter 10 Introduction to Probability

Chapter 4 Introduction to Probability
-in the SR (short-run) events may be viewed as unpredictable, but in the LR (long-run)
patterns tend to emerge and we can view a predictable pattern. This is what probability
attempts to capture.
A. Probability in General
1. Probability – a numerical measure of the likelihood that an event will occur. It is
obtained by measuring a very large number of outcomes.
Notation: P ( event ) =
Properties:
(a) Takes on a value from 0 to 1 0 means it cannot occur and 1 (or 100% ) means it will
occur absolutely and is therefore not by chance.
(b) As we go from 0 to 1 the likelihood of an even occurring is increasing
(c) If something has a probability of 0.5 it is just as likely to occur as to not occur.
2. Experiment (pertaining to probability) – process that generates well-defined
probabilities. On any one repetition of an experiment any outcome that is in the sample
space may occur.
3. Sample Space – S- set of all possible experimental outcomes. One sample point is one
experimental outcome. Generally denoted in some sort of bracket  {} or [ ].
Ex: roll of a die  experiment
Sample space  S = { 1, 2, 3, 4, 5, and 6 } or all possible outcomes that may occur
Sample point  on individual outcome…say 1
Probability  since all are equally likely P (event) = 1/6 or for an individual outcome
we may write P (rolling a 1) = 1/6
4. Random – when an individual outcomes or outcomes are uncertain and there is
‘regular’ or measurable distribution of outcomes in a large number of repetitions.
5. Multiple Step Experiments – when there is more than one step in the experiment.
Before we considered a single step probability, now we look at what happens when there
are multiple outcomes we are concerned with.
Ex: flipping two coins or two die instead of one.
a. Counting Rule for Multiple Steps – If an experiment can be described as a sequence of
k steps with n possible outcomes with n1 possible outcomes on the first step, n2 possible
outcomes on the second step and so one, then the total number of experimental outcomes
is given by the expression:
Total outcomes = n1* n2*…….* nk
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Example1: Consider flipping two coins. Then we identify the following items;
k = 2 for two steps (i.e. two coins)
n1 , n2 = 2 for the two outcomes for each (heads and tails)
Total outcomes = 2 * 2 = 4
Example 2: Consider the flip of a coin and the roll of a die. Then we identify the
following items;
k = 2 for two steps (i.e. one coin and one die roll)
n1 = 2 for the two outcomes (heads and tails)
n2 = 6 for the six outcomes [ 1, 2, 3, 4, 5, and 6 ]
Total outcomes = 2 * 6 = 12
Note: we still have not assigned probabilities to the individual outcomes that may occur
from performing the next experiment. We are simply identifying the total number of
outcomes that may occur given certain conditions.
b. Combinations – used when we want to count the number of experimental outcomes
from an experiment that involves selecting n objects from a total population of N objects.
C nN  ( nN ) = N! / n! (N-n)!
Where N! = (N)(N-1)(N-2)….(3)(2)(1)
n! = (n)(n-1)(n-2)….(3)(2)(1)
0!=1
n<N
example: Consider we check the quality of meat at 3 supermarkets out of a possible 10.
What are all the possible combinations we could pick Then N = 10 and n = 3.
C nN  C310  ( 10
3 ) = 10! / 3! (10-3)! = 10! / 3!7! 
10*9*8*7*6*5*4*3*2*1 / [3*2*1]*[7*6*5*4*3*2*1] = 10*9*8 / 6 = 120
c. Permutations – This is very much like combinations but now order matters. So even if
the outcomes have the same objects, if the order is different then we consider it a
different outcome (i.e. if we picked Paul, Sharon, and James in that order and on another
turn in the same experiment picked Sharon, Paul, and James…since the order is different
even though the outcome has the same results it is considered different from the first).
Since this is the case we would expect to have more outcomes and if we look at the
formula we see that.
Mathematical expression: PnN  n!( nN ) = N! / (N-n)!
Where N! = (N)(N-1)(N-2)….(3)(2)(1)
n! = (n)(n-1)(n-2)….(3)(2)(1)
0!=1
n<N
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example: Consider the same example, but now we check the quality of meat at 3
supermarkets out of a possible 10 and order matters. We still have N = 10 and n = 3.
PnN  P310  3!( 10
3 ) = 10! / (10-3)! = 10! / 7! 
10*9*8*7*6*5*4*3*2*1 / 7*6*5*4*3*2*1] = 10*9*8 = 720
6. Assigning Probabilities to Outcomes
-now we want to actually see how likely a specific outcome is. To do that we first have
some basic requirements to assign probabilities which are:
(a) The probability assigned must be between 0 and 1. So if we let Ei denote the
experimental outcome, then P(Ei) is the probability of the outcome and it must be
0 ≤ P(Ei) ≤ 1 for all i
(b) the sum of all probabilities for all the experimental outcomes (i = 1….n) must be
equal to 1. Written another way we say Σ P(Ei) = P(E1) + P(E2) + ….+P(En) = 1
-If all the outcomes are equally likely then the probability of one event is 1/n
-if the outcomes are not equally likely then we take the number of occurrences of one
outcome, add them up and divide by the total number of outcomes
P ( Event ) = total # occurrences of that event / Total outcomes of all events
-it is possible to do this subjectively and express the probability (still between 0 and 1) as
a belief, but it is only as reliable as the knowledge of the person stating the belief. Called
a personal probability or subjective probability.
7. Probability Model – a mathematical description of the sample space and a way of
assigning the probability to the events in the sample space.
B. Events, Their Probabilities, & Probability Rules
1. Event – a collection of sample points. This is akin to what we call a sample space.
2. Probability of an Event – the probability of an event is the sum of all probabilities of
the sample points that are in the event.
Ex: Consider the rolling of a die. The event is the rolling of a die and its outcomes.
The probability of the event is P(rolling 1) + P (rolling 2)+…+P(rolling 6) = 1
3. Complement of an Event A – given an event A, the complement is all the sample
points not in A.
Mathematically:
i)
P(A) = 1 – P (Ac) or
ii)
P(A) + P (Ac) = 1
4. Venn Diagram – diagram that illustrates the concept of complements (and other
related probability displays).
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Note: The entire diagram is the sample space.
Ac
Event A
5. Union of two Events – the union of two events A and B is the event containing all the
sample points of both A and B.
Mathematically: A  B
Graphically:
A
B
Note: You don’t double count the space that is common between A and B. This will be
noted later.
6. Intersection of two Events – This is the shade regions above. It is what is common
among both A and B.
Mathematically: A  B
7. General Addition Law – P(A  B ) = P(A) + P(B) – P(A  B)
Ex: suppose that we have 1000 workers total and we want to know what the
P( Woman and Over 50) = P ( Woman  Over 50)?
If we are given that there are 300 women and 500 workers are over 50 we don’t want to
double count the over 50 workers that are also women. So,
P (Woman) = 0.3
P(Over 50) = 0.5
If we are given the number of female workers over 50 to be 100, then
P ( Woman  Over 50) = 0.3 + 0.5 – 0.1 = 0.7 or 70%
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8. Mutually Exclusive – two events are said to be mutually exclusive if there are no
sample points in common. If this is the case then the additive law reduces to
P(A  B ) = P(A) + P(B) since there is no intersection that would double up the count on
certain data points.
Graphically:
A
B
9. Summary of Probability Rules
a) The probability of any event must lie somewhere between 0 and 1.
b) The probability of the entire sample space must sum to 1.
c) Two events are said to be disjoint if there are no outcomes in common.
d) The probability of A complement is 1 – P(A)
10. Conditional Probability – the probability of an event occurring given that another
event has already occurred. This is in essence an update of your information and
modifies what sample points are now of interest.
Mathematically:
a. P (A|B) = P(A  B) / P(B)
b. P (B|A) = P(A  B) / P(A)
Ex: Suppose we are given the following table and you are asked the following questions.
Age \ Marital Status Single
Married
Totals
Under 30
77
14
91
30 or Over
28
21
49
Totals
105
35
150
Find the probability of being single given that the subject is under 30. This is conditional
since we now focus on only the under 30 group.
P (Single|Under30) = P(single  under 30) / P(under 30) = (77/150) / (91/150) = 77 / 91
= 0.8462 or 84.6%
11. Independent Events – when one event is not influenced by another event (this is not
mutually exclusive  This is something that is commonly missed).
Conditions for Independence:
(a) P (A|B) = P(A)
(b) P (B|A) = P(B)
(c) P (A  B) = P(A)*P(B)
-Otherwise the events are said to be dependent.
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Ex: Lets go back to our table above and see if marital status and age are independent of
each other for one of the categories.
Consider P (Single|Under30) = 84.6%. Is it the case the P(Single) also equals 84.6%
making the events of being single and under 30 independent of one another?
P(Single) = 77 / 150 51.3%. So in this case the events are dependent, but if the
percentages were equal they would be independent.
12. General Rule for Multiplication – Gives us the probability that two events occur;
used in conjunction with conditional probabilities.
Mathematically:
a) P (A  B) = P(A)* P(B|A)
b) P (A  B) = P(B)* P(A|B)
13. Baye’s Theorem – provides a means for updating initial/prior probabilities when
there is updated information available. These updated priors are called posterior
probabilities. This can really only be done for mutually exclusive events.
Mathematically: P (Ai|B) = P (Ai) P(B|Ai) / Σ P (Ai) P(B|Ai)
Note: The denominator is just the total probability formula. This formula in general
shows us how to start with initial and conditional probabilities and gives how to update
them to proper posterior probabilities.
14. Tree Diagrams – it is a diagram that looks like the branches of a tree. It shows the
events that occur along with their respective probabilities along the branches. The
multiplication rule can be used to find the overall probability of events by simply
multiplying the probability of events along an entire branch
P(Head) = 0.5
Ex:
Probabilities
Head
(.5)(.5) = 0.25
P(Head) = 0.5
Head
P(Tail) = 0.5
Flip a Coin
P(Tail) = 0.5
P(Head) = 0.5
Tail
P(Tail) = 0.5
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Tail
Head
Tail
(.5)(.5) = 0.25
(.5)(.5) = 0.25
(.5)(.5) = 0.25
C. Continuous vs. Discrete Distributions
1. Random Variable – a variable whose value is a numerical outcome is a random
phenomenon. Can be either discrete or continuous.
a. Discrete random variable – a random variable that takes on whole numbers or integers.
b. Continuous random variable – when the random variable can take on any value in an
interval; not just whole numbers.
2. Discrete Probability Model – This is a probability model that has a finite sample
space and also takes on a finite value or infinite sequence of values.
3. Continuous Probability Model – this is a probability model that has an infinite
sample space. We cannot assess the probability of an outcome at a point in this case 
the reasoning is the probability of something at one point under a distribution is zero
since the area would be 0.
-probability in this case must be measured using a range of values and is the area under
the curve in that range.
Ex: a uniform distribution or normal distributions are good examples.
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