Probability

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Probability StatisticsLecture 1

• Michael Partensky

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Topics of Discussion (1)

• The Laws of Chance Are they possible?
• Gambling its serious
• On the shoulders of Giants
• Collecting the data
• Random experiments and events.
• Sample space, sample sets.
• Operations on sample sets.
• Frequencies of events
• The first definition of probability (P)
• Axioms of probability
• Frequency interpretation of P
• General properties of P (derivation from Axioms).

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Topics of Discussion (2)

• Random variables and distribution functions.
• Discrete and continuous variables. Examples.
• Distribution function for discrete variables.
• Continuous distributions.

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On the shoulders of Giants
Blaise Pascal, Frenh mathematician and
philospher (1623-1662)
Pierre de Fermat, French mathematician
(1601-1655)

Christiaan Huygens, Dutch astronomert,
mathematician, physicist (1629-1695)
Jacob Bernoulli,Swiss mathematician(1654-1705)
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Not only only from their achievements, but also
from errors
D'Alembert argued, that if two coins are tossed,
there are three possible cases, namely(1) both
heads, (2) both tails, (3) a head and a tail. So
he concluded that the probability of  " a head
and a tail " is 1/3. If he had figured that this
probability has something to do with the
experimental frequency of the occurrence of the
event, he might have changed his mind after
tossing two coins more than few times.
Apparently, he never did so. Why? We do not
know. Was he right? Think!
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A famous statistician would never travel by
airplane, because she had studied air travel and
estimated the probability of there being a bomb
on any given flight was 1 in a million, and she
was not prepared to accept these odds. One day a
colleague met her at a conference far from home.
"How did you get here, by train?" "No, I flew"
"What about the possibility of a bomb?" "Well,
I began thinking that if the odds of one bomb are
1million, then the odds of TWO bombs are
(1/1,000,000) x (1/1,000,000) 10-12. This is a
very, very small probability, which I can accept.
So, now I bring my own bomb along!"
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We will be dealing a lot with the paradoxes of
probability. Here is a cute brain-stretcher.
Adventures on the Moscow Subway Boris commutes
to college by the Moscow subway which runs in a
circle. The school happens to be at the point on
the circle that is exactly opposite to where
Boris boards the train. The trains run in both
directions, and the schedule is very regular. The
time intervals for trains in both directions are
equal for instance, if there is an hour between
the arrival of clockwise (cw) trains, there is
also an hour between the arrival of
counterclockwise trains. Boris observed, however,
that he caught the cw trains much more often than
the ccw trains., despite the fact that his
schedule was irregular and he arrived at
theStation at random times. Can you explain this?
T-station
School
From The chicken from Minks by Y.B. Chernyak
and R.M. Rose
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Random experiments

• Term "random experiment" is used to describe any
  action whose outcome is not known in advance.
  Here are some examples of experiments dealing
  with statistical data
• Tossing a coin
• Counting how many times a certain word or a
  combination of words appears in the text of the
  King Lear or in a text of Confucius
• Counting how many Japanese sedans passed the
  Washington Bridge between 12 and 12.30 p.m.
• counting occurrences of a certain combination of
  amino acids in a protein database.
• pulling a card from the deck

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Sample spaces, sample sets and events
The sample space of a random experiment is a set
that includes all possible outcomes of the
experiment. For example, if the experiment is to
throw a die and record the outcome, the sample
space is
the set of possible outcomes. describes an
event that always occurs.Each outcome is
represented by a sample point in the sample
space. There is more than one way to view and
experiment, so an experiment may have more than
one associated sample space. For example,
suppose you draw one card from a deck. Here are
some sample spaces. Sample space 1 (the most
common) The space consists of 52 outcomes, 1 for
each card. Sample space 2 The space consists of
just two outcomes, black and red, Sample space 3
This space consists of 13 outcomes, namely
2,3,4, 10,J,Q,K,A. Sample space 4. This space
consists of two outcomes, picture and
non-picture. In tossing a die, one sample space
is 1,2,3,4,5,6, while two others are odd,
even and less then 3.5, more then 3.5
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The examples above describe the discrete sample
spaces. Intuitively, you all understand what it
means although a precise mathematical definition
would be cumbersome. We will discuss the
continuous sample spaces a little later. The
examples can be represented by various physical
variables describing a population, such as
weight, height, concentration etc.
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• Race between seven horses,1,2,3,4,5,6,7.
• Experiment 1 determining the winner.
• Experiment 2 determining the order of finish
• Describe the sample spaces

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Events
Certain subsets of the sample space of an
experiment are referred to as events. An event is
a set of outcomes of the experiment. This
includes the null (empty) set of outcomes and
the set of all outcomes. Each time the experiment
is run, a given event A either occurs, if the
outcome of the experiment is an element of A, or
does not occur, if the outcome of the experiment
is not an element of A. What to consider an
event is decided by the experimentalist.
Examples. In the Sample space 1 (see above) of
52 cards the following events can be
considered among others A drawing a king, B
drawing Q or A, C drawing a non-picture, D
drawing a black. For example, event D consists
of 26 points. D is also an event in sample space
2 (consisting of 1 point). It is not an event in
sample spaces 3 and 4. Similarly, A (king) is
an event in sample spaces 1 and 3 but not in 2
and 4.
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The examples of sample spaces and events
Example 1.1 Flip two coins. Try to figure out
what is the sample space for this experiment How
many simple events does it contain?
The answer could be described by a following
table
1 \ 2 H T
H HH HT
T TH TT
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The examples of sample spaces and events
Example 1.2 Role two dice. For convenience we
assume that one is red and the other is green (we
often use this trick which hints that we can
tell between two objects which is which). The
following table describes
1 2 3 4 5 6
1 11 12 13 14 15 16
2 21 22 23 24 25 26
3 31 32 33 34 35 36
4 41 42 43 44 45 46
5 51 52 53 54 55 56
6 61 62 63 64 65 66
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Example 1.2 (continued) . Any event corresponds
to some collection of the cells of this table. We
showed here four different events (colored blue,
red, green and gray) . Try to describe them in
plain English
1 2 3 4 5 6
1 11 12 13 14 15 16
2 21 22 23 24 25 26
3 31 32 33 34 35 36
4 41 42 43 44 45 46
5 51 52 53 54 55 56
6 61 62 63 64 65 66
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Example 1.2 (continued) . And what about the
following events?Which of these event (red,
orange, purple, etc) occurs more often?
1 2 3 4 5 6
1 11 12 13 14 15 16
2 21 22 23 24 25 26
3 31 32 33 34 35 36
4 41 42 43 44 45 46
5 51 52 53 54 55 56
6 61 62 63 64 65 66
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Sample spaces and events

• Example 1.3 An experiment consists of drawing
  two numbered balls from the box of balls numbered
  from 1 to 5. Describe the sample space if
• The first ball is not replaced before the second
  is drawn.
• The first ball is replaced before the second is
  drawn.
• Example 1.4 Flip three coins. Show the sample
  space. What is the total number of all possible
  outcomes?

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Composite events
Quite often we are dealing with composite events.
Example. We study a group of students, picking
them at random and considering the following
events A A student is female, BA student
is male, C A student has blue eyes, DA
student was born in California. After this
information was collected and the probabilities
of A. B, C and D were determined, we decided to
find the probabilities of some other events U
Student is female with blue eyes, VStudent is
male, or has blue eyes and was not born in
California, etc. The latter questions are
dealing with the events that are composed from
the atomic events A, B, C and D. To describe
them, it is convenient to use a language of the
Set Theory. Warning please, dont be scared. We
are not going to be too theoretical. The new
symbols will be introduced for our convenience.
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Inclusion A is a subset of BOccurrence of A
implies the occurrence of B. Example B
2,3,5,6, A2,6
A
B
A?B
Intersection belongs to A and B
Examples (1) B 2,3,4,6, A1,2,5,6
2,6 (2) AFemale students,
BStudents having blue eyes,
Female students with blue eyes.
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Operations on sample sets (1)
The empty set is the event with no
outcomes. The events are disjoined if they do
not have outcomes in common Example
A1,3,4, B 2,6
The union of events A and B, is the
event that A or B or both occur. Example A
Male student, B Having blue eyes,
Students who are either male or have
blue eyes (or both)
Complementary (opposite) eventsAc is the
compliment to A if and
W
A
Ac
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The first definition of probability

• We have introduced some important concepts
• Experiments,
• Outcomes,
• Sample space,
• Random variables.
• These concepts will come to life after we
  introduce another crucial concept- probability,
  which is a way of measuring the chance.
• A probability is a way of assigning numbers to
  events that satisfies following conditions or

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The Axioms of Probability

• (1) For any event A, 0 P(A) 1.
  (1.1)
• (2) If is the sample space, then  P( )1
  (1.2)
• (3) For a sequence of disjoined (incompatible)
  events Ai (finite or infinite),

In other words, the probability for a set of
disjoined events equals the sum of individual
probabilities
We will introduce here another important
property, although it is not an Axiom and will be
justified later in the lecture devoted to the
analysis of conditional probability
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(4) If A and B are independent, then
In other words, for any number of independent
events, N,
Incompatible (disjoined) and independent events

• These new concepts introduced above require some
  explanation.
• Two events are said to be incompatible or
  disjoined if they can not occur together.
• Two events are said to be independent if they
  have nothing to do with each other.

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• Examples
• (Disjoined) Jack can arrive to Boston either on
  Monday (M) or on Wednesday (W). These events
  can not occur together (he can not arrive on
  Monday and on Wednesday). Therefore the events
  are disjoined. If P(M)0.72 and P(W)0.2,
  thenP(M or W)0.92 (there is still a chance that
  Jack wont come to Boston at all).
• (Independent) A It will be sunny today,
  BCeltic will win the game tomorrow. What si
  the probability that both events will occur
  simultaneously?P(A and B)P(A) P(B).
• Please, offer some more examples of both kinds.

You can find a very useful and provocative
discussion of these concepts and of the
probability in general in the book Chance and
Chaos by David Ruelle (Princeton Sci. Lib.)
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Frequency interpretation of probability

• If we repeat an experiment a large number of
  times, then
• the fraction of times the event A occurs will be
  close to P(A).
• In other words, if N(A,n) is the number of times
  that event occurs
• in the first n trials, than
• It's easy to prove that defined this way, P(A)
  satisfies conditions
• (1) and (2) (Try to prove it).
• Hint the property (3) follows from

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Some other properties of probability. Deduction
from the axioms 1-3
Try to prove (a) and (b). The property (c) is
harder to prove formally, although intuitively it
is quite clear. Summing up the areas of A and B,
one counts their intersection twice. The
probability is kind-a proportional to the area.
Therefore, one of the occurrences of AB should be
removed.
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Comment In many cases it is easier to find a
probability of a compliment event Ac rather then
A itself. It such cases, the rule (1.7) can be
very helpful. Example Toss two dice. Find the
probability that they show different values, for
example (4,6) and (2,3) but not (2,2). You can
count favorable outcomes directly, or better
still, by (3) P(non-matching dice) 1
P(matching dice) 1 6/36 5/6. We will be
using this trick quite often.
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• Example the classical birthday problem.
• Find a probability that in a group of n people
  at least two will have the same birthday.
• You will solve a version of this problem at home.
• Lets discuss here a simpler problem.
• N people arrive to a vacation place during the
  same week. Each one chooses randomly his/her
  arrival day. What is the probability that at
  least two people arrive on the same day?
• Lets discuss it together (working in groups)
• What is the complimentary event?
• Consider first N2,
• N8.
• Now let us check N3.
• The general case

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Random Variables and the Distribution Function.

• The simplest experiments are flipping coins and
  throwing dice. Before we can say anything about
  the probabilities of their various outcomes (such
  as "Getting an even number" on the die, or
  "Getting 3 heads in 5 consecutive experiments
  with a coin") we need to make a reasonable guess
  about the probabilities of the elementary events
  (getting H or T for a coin , or one of six faces
  for a die). For instance, we can assume (as we
  usually do) that a die is perfectly balanced and
  all faces are equivalent. Then , the probability
  of any number equals 1/6.
• As we will see, it can be described in terms of
  distribution (because it distributes
  probabilities between different outcomes)
  function.
• The term function, however, implies some
  arguments (or variables). It would be
  inconvenient to use a function of faces, or a
  functions of Heads or Tails. Thats why we will
  introduce a general term for describing various
  random outcomes.

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Random Variables

• We now introduce a new term
• Instead of saying that the possible outcomes are
  1,2,3,4,5 or 6, we say that random variable X
  can take values 1,2,3,4,5,6.A random variable
  is an expression whose value is the outcome of a
  particular experiment.
• The random variables can be either discrete or
  continuous.
• Its a convention to use the upper case letters
  (X,Y) for the names of the random variables and
  the lower case letters (x,y) for their possible
  particular values.

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Examples of random variables

• Discrete (you name !)
• Continuous
• For instance, weights or a heights of people
  chosen randomly, amount of water or electricity
  used during a day, speed of cars passing an
  intersection.
• Please, add a few more examples.

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The Probability Function for discrete random
variables

• We assigned a probability 1/6 to each face of the
  dice. In the same manner, we should assign a
  probability 1/2 to the sides of a coin.
• What we did could be described as distributing
  the values of probability between different
  elementary events
• P(Xxk)p(xk),
  k1,2,
  (1.9)It is convenient to introduce the
  probability function p(x) P(Xx)p(x)
  (1.10)
• In other words, the probability of a random
  variable X taking a particular value x
• Is called the probability function.
• For xxk (1.10) reduces to (1.9) while for other
  values of x, p(x)0.

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The probability function should satisfy the
following equations
Example Suppose that a coin is tossed twice, so
that the sample space is ?HH,HT,TH,TT. Let X
represent a number of heads that can come up.
Find the probability function p(x). Assuming
that the coin is fair, we have P(HH)1/4,
P(HT)P(TH)1/4, P(TT)1/4Then,
P(X0)P(TT)1/4 P(X1)P(HT?TH)1/41/41/2.
P(X2) ¼.
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The probability function is thus given by the
table
x 0 1 2
p(x) 1/4 1/2 1/4
The probability function p(x) is related to the
probability density function (PDF) f(x)
introduced in the next section for the continuous
random variables. For those familiar with the
concept of ?-function, this relation can be
presented as
All others can simply ignore this formula.
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Uniform and non-uniform distributions.
If all the outcomes of an experiment are equally
probable, the corresponding probability function
is called uniform. If the contrary is true, the
probability function is non-uniform. Example On
the face of a die with 6 dots, 5 dots are filled,
so that only the central one is left. What is the
probability function for this case? The value X1
will occur in average 2 times more often than in
the balanced die. As a result, p(1)1/3,
p(2)p(3)p(4)p(5) 1/6 . Working
together TryIt Suppose we pick a letter at
random from the word TENNESSEE. What is the
sample space and what is the probability function
for the outcomes? Challenge For two dice
experiment, find the probability function for
X Sum of two throws.
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Continuous distribution (preliminary remarks)
Suppose that the circle has a unit circumference
(we simply use units in which 2 Pi R1).
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Continuous distribution. Probability density
function (PDF).
Suppose that every point on the circle is labeled
by its distance x from some reference point x0.
The experiment consists of spinning the pointer
and recording the label of the point at the tip
of the pointer. Let X is the corresponding random
variable. The sample space is the interval 0,1).
Suppose that all values of X are equally
possible. We wish to describe it in terms of
probability. If it was a discrete variable (such
as a dice), we would simply assign to every
outcome a fixed value of probability to all
outcomes.. p(xi)const. However, for a continuous
variable we must assign to each outcome a
probability p(x )0. Otherwise, we would not be
able to fulfill the requirement 1.12. Something
is obviously wrong!
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Continuous distribution and the probability
density function
Dealing with the infinitesimal numbers is a
tricky business indeed. Those who studied
calculus aware of this.
A random variable X is said to have a continuous
distribution with density function f(x) if for
all a? b we have
The analogs of Eqs. 1.12 and 1.13 for the
continuous distributions would be
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P(E) is a probability that X belongs to E.
f(x)
P(altXltb)
a
b
Geometrically, P(altXltb) is the area under the
curve f(x) between a and b.
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Examples 1. The uniform distribution on
(a,b) We are picking a value at random from
(a,b).
By direct integration you can verify that (1.18)
satisfies the condition (1.16). We can now find
PDF which describes the experiment with the
spinner in which case b-a2?
The probability that the arrow will stop in the
rage between ? and ? ?? equals ??/2?.
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2. The exponential distribution
Those who know how to integrate can verify that
(1.19) satisfies (1.16) (the total area under the
curve f(x) equals 1. Note In Matematica, the
integral of a function fx (notice that
rather than () is used) can be found
as Integratefx,x,x1,x2 , ShiftEnter. Here
x1 and x2 are the limits of integration. Please,
practice with this at home.
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3. The standard normal distribution
Using Mathematica check that this (corrected)
formula satisfies the second Axiom of probability
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Self-Test

• 1. For what number of students n the
  probability of at least two having the same
  birthday reaches 0.7?
• A card is drawn at random from an ordinary deck
  of 52 playing cards. Describe the sample space if
  consideration of suits (hearts, spades, etc) (a)
  is not, (b) is, taken in the account.
• Referring to the previous problem, let A"a king
  is drawn" and B"a club is drawn". Describe the
  events (a)AUB, (b) AB, (c) A?Bc (d) Ac ? Bc (d)
  (AB) ?(ABc)
• Describe in words the events specified by the
  following subsets of
  HHH,HHT,HTH,HTT,THH,THT,TTH,TTT
• E HHH,HHT,HTH,HTT (b) E HHH,TTT (c) E
  HHT,HTH,THH
• (d) E HHT,HTH,HTT,THH,THT,TTH,TTT
• 5. A die is loaded in such a way that the
  probability of each face turning up is
  proportional to the number of dots on that face
  (for instance, six is three times as probable as
  two). What is the probability of getting an even
  number in one throw?

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6. Let A and B be events such that
What is the P(AUB)? 7. A student must choose one
of the subjects art, geology, or psychology as
an elective. She is equally likely to choose art
or psychology, and twice as likely to choose
geology. What are the respective probabilities
for each of three subjects?