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# Probability and Statistics. Lecture 2. Chapter 3

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Probability and StatisticsLecture 2

Chapter 3

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Where are we going?We have a population (a basket) that we cannot examine but

we want to learn something about it - so we will take a

sample - preferably it will be a random sample.

We will use the sample to estimate the things we wanted to

know about the population - we will use the sample results

to test theories about the population and make some

decisions.

Since the sample is just a part of the population there will be

some uncertainty about the estimates and decisions we

make. To measure and quantify that uncertainty we turn to

PROBABILITY!

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The word probability is a commonly used term that relates to the chancethat a particular event will occur when some experiment is performed.

An experiment for us is any process that produces an observation, or

outcome.

We are often concerned with an experiment whose outcome is not

predictable, with certainty, in advance.

Even though the outcome of the experiment will not be known in advance,

we will suppose that the set of all possible outcomes is known. This set of

all possible outcomes of the experiment is called the sample space and is

denoted by S.

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Many decisions that we make involve uncertainty and the evaluation of probabilities.Interpretations of Probability

Example: Roll a fair die possible outcomes = {

}

Before you roll the die do you know which one will occur?

What is the probability that the outcome will be a ‘4’? __________ Why?

A few ways to think about PROBABILITY:

(1)

Personal or Subjective Probability

P(A) = the degree to which a given individual believes that the event A will happen.

(2)

Long term relative frequency

P(A) = proportion of times ‘A’ occurs if the random experiment (circumstance) is repeated many,

many times.

(3) Basket Model

P(A) = proportion of

balls in the basket

that have an ‘A’ on

them.

10 balls in the basket: 3

blue and 7 white

One ball will be

selected at random.

What is P(blue)?

______________

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Some examples of experiments and their sample spaces are as follows.(a) If the outcome of the experiment is the gender of a child, then S = {g, b} where

outcome g means that the child is a girl and b that it is a boy.

(b) If the experiment consists of flipping two coins and noting whether they land heads

or tails, then

S = {(H,H), (H, T), (T,H), (T, T)}

(d) Consider an experiment that consists of rolling two six-sided dice and noting the

sides facing up. If we let (i, j ) denote the outcome in

which die 1 has value i and die 2 has value j , then the sample space of this experiment

is

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Any set of outcomes of the experiment is called an event.That is, an event is a subset of the sample space. Events will be denoted by the capital

letters A,B,C, and so on.

In Example 4.1(d), if

A = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}

then A is the event that the sum of the dice is 7.

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A graphical representation of events that isvery useful is the Venn diagram.

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For any two events E and F, if all of theoutcomes in E are also in F, then we say

that E is contained in F and write E ⊂ F.

Thus if E ⊂ F , then the occurrence of E

necessarily implies the occurrence of F.

If E ⊂ F and F ⊂ E, then we say that E

and F are equal (or identical) and

we write E = F

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Proof:## 14.

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SAMPLE SPACES HAVING EQUALLY LIKELY OUTCOMESFor a large number of experiments, it is natural to assume that

each point in the sample space is equally likely to occur.

BASIC PRINCIPLE OF COUNTING

Suppose that two experiments are to be performed. Then if experiment 1 can result in

any one of m possible outcomes and if, for each outcome of experiment 1, there are n

possible outcomes of experiment 2, then together there are mn possible outcomes of the

two experiments

Example: two dice rolled, two coins tossed.

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Generalized Basic Principle of CountingIf r experiments that are to be performed are such that the first one may result in

any of n1 possible outcomes, and if for each of these n1 possible outcomes there

are n2 possible outcomes of the second experiment, and if for each of the

possible outcomes of the first two experiments there are n3 possible outcomes of

the third experiment, and if, . . . , then there are a total of n1 · n2 · · · nr possible

outcomes of the r experiments.

n! = n(n−1)(n−2) · · · 3 · 2 · 1

Example 2.17: How many even four-digit numbers can be formed from the

digits

0, 1, 2, 5, 6, and 9 if each digit can be used only once?

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As an illustration of this, let us determine the number of different waysn distinct objects can be arranged in a linear order.

For instance, how many different ordered arrangements

of the letters a, b, c are possible? By direct enumeration we see that

there are 6; namely, abc, acb, bac, bca, cab, cba. Each one of these

ordered arrangements is known as a permutation.

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Suppose now that we are interested in determining the number ofdifferent groups of r objects that could be formed from a total of n

objects.

Suppose now that we are interested in determining the number of

different groups of r objects that could be formed from a total of n

objects.

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EXAMPLE 3.5d A committee of size 5 is to be selected from a group of 6 men and9 women.

If the selection is made randomly, what is the probability that the committee

consists of

3 men and 2 women?

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EXAMPLE 3.5fA basketball team consists of 6 black and 6 white players. The players are to be paired in

groups of two for the purpose of determining roommates.

If the pairings are done at random, what is the probability that none of the black players

will have a white roommate?

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CONDITIONAL PROBABILITYIn this section, we introduce one of the most important concepts in all of

probability theory —that of conditional probability. Its importance is twofold.

In the first place, we are often interested in calculating probabilities when some

partial information concerning the result of the experiment is available,

/or in recalculating them in light of additional information.

In such situations, the desired probabilities are conditional ones.

Second, as a kind of a bonus, it often turns out that the easiest way to compute

the probability of an event is to first “condition” on the occurrence or

nonoccurrence of a secondary event.

the conditional probability of E given that F has occurred, and is

denoted by

P(E|F )

=(Number of points in EF) /

Number of points in F

Restatement of the definition

P(EF)=P(F)* P(E|F)

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Example. A class consisting of 4 graduate and 12 undergraduate students israndomly divided into 4 groups of 4.

What is the probability that each group has a graduate student?

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Total probability## 28.

You take part in a chess tournament where your probability of winning is 0,3against half the players (type 1), 0,4 against a ¼ of the players (type 2), and 0,5

against the remaining players (type 3) . You play a game against a randomly

chosen opponent. What is the probability of winning?

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REVERSEP(A|B) vc P(B|A)

In Chess problem - Suppose that you win. What is the

probability that you had an opponent of type 1?

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EXAMPLE 3.7c In answering a question on a multiple-choice test, a studenteither knows the answer or she guesses. Let p be the probability that she

knows the answer and 1−p the probability that she guesses. Assume that a

student who guesses at the answer will be correct with probability 1/m,

where m is the number of multiple-choice alternatives.

What is the conditional probability that a student knew the answer to a

question given that she answered it correctly?

Bayes Rule

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Suppose we have 3 cards identical in form except that both sides of the 1stcard are colored red; both sides of the 2nd card are colored black; 3rd card –

one side red, other side is black

The three cards mixed up in hat, and one card is

randomly selected and put down on the ground.

If the upper side of the chosen card is red, what

is the probability that the other side is colored

black?

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Using conditional probability for modelingIf an aircraft is present in a certain region, a radar correctly registers its

presence with probability 0,99.

If it is not present, the radar falsely registers an aircraft presence with

probability 0,1.

Assume that aircraft is present with probability p=0,05.

Find:

a) Probability of false alarm (false indication of presence)

b) Probability of missed detection (i.e. nothing registered even though is

present)

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IndependenceWe introduced conditional probability to catch the partial information that event

B provides about event A

P(A|B)

If occurrence of event B has no influence to A

P(A|B)= P(A)

we say that A is independent of B

P(AB)= P(A) P(B)

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Theorem. If E and F are independent , thenE and