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**Chapter 2 Concepts of Prob. Theory**

Random Experiment: an experiment in which outcome varies in an unpredictable fashion when the experiment is repeated under the same condition Specified by: 1. An experimental procedure 2. One or more measurements or observations tch-prob

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EXAMPLE 2.1 : Experiment E1 : Select a ball form an urn containing balls numbered 1 to 50. Note the number of the ball. Experiment E2 : Select a ball form an urn containing balls numbered 1 to 4. Suppose that balls 1 and 2 are black and that balls 3 and 4 are white. Note number and color of the ball you select. Experiment E3 : Toss a coin three times and note the sequence of heads/ tails. Experiment E4 : Toss a coin three times and note the number of heads. Experiment E6 : A block of information is transmitted repeatedly over a noisy channel until an error-free block arrives at the receiver. Experiment E7 : Pick a number at random between zero and one. Experiment E12 : Pick two numbers at random between zero and one. Experiment E13 : Pick a number X at random between zero and one, then pick a number Y at random between zero and X. tch-prob

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**Discussions Compare E3 and E4: same procedure, different observations**

A random experiment with multiple measurements or observations: E2, E3, E12, E13 sequential experiment consists of multiple subexperiments: E3, E4, E6, E12, E13 dependent subexperiments: E13 tch-prob

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**Sample Space Outcome, sample point:**

one and only one per experiment mutually exclusive, cannot occur simultaneously Sample space: set of all possible outcomes Example 2.2: sample spaces number of outcomes: Finite countably infinite S6 unaccountably infinite S7 Discrete sample space Zeta is the symbol name Continuous Sample space tch-prob

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EXAMPLE 2.2 : The sample spaces corresponding to the experiments in Example 2.1 are given below using set notation : tch-prob

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**elementary event: event of a single outcome**

Event: a subset of S, a collection of outcomes that satisfy certain conditions Certain event: S null event: elementary event: event of a single outcome Example 2.3: A2: The ball is white and even-numbered A4: The number of heads equals the number of tails A7: The number selected is nonnegative tch-prob

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**A B Set Operations Union Intersection mutually exclusive Complement**

imply equal A=B - Commutative properties of set operations - Associative properties of set operations - Distributive properties of set operations - De Morgan’s Rules Venn diagrams: Fig. 2.2 A B tch-prob

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**Example 2.5: Configuration of a three-component system **

a. series all three b. parallel at least one of three c. two-out-of-three C1 C2 C3 C1 C2 C3 tch-prob

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**2.2 The Axioms of Probability**

A probability law for the experiment E is a rule that assigns to each event a number P(A), called the probability of A, that satisfies the following axioms: Axiom I: nonnegative Axiom II: P(S)= total=1 Axiom III: If , then Axiom III’: If for all , then Axiom: a statement accepted as true as the basis for argument or inference an established rule or principle or a self-evident truth A set of consistency rules that any valid probability assignment must satisfy. tch-prob

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**Corollary 4. are pairwise mutually exclusive, then for**

pf: Corollary 2. pf: from Cor.1, Corollary 3. pf: Let A=S, in Cor.1. Corollary are pairwise mutually exclusive, then for Corollary: a proposition inferred immediately from a proved proposition with little or no additional proof tch-prob

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Corollary 5. pf: Corollary 6. Corollary 7. If , then pf. tch-prob

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**Discrete Sample Spaces find the prob.of elementary events; **

Axioms + corollaries provide rules for computing the probability of certain events in terms of other events. However, we still need initial probability assignment Discrete Sample Spaces find the prob.of elementary events; all distinct elementary events are mutually exclusive, Example 2.6., 2.7 2. Continuous Sample Spaces assign prob. to intervals of the real line or rectangular regions in the plane Example 2.11 tch-prob

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**Discrete Sample Spaces **

First, suppose that the sample space is finite, and is given by (by corollary ?) If S is countably infinite, then Axiom Ⅲ’ implies that the probability of an event such as is given by tch-prob

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**The probability of any event that consists of k outcomes, say , is**

If the sample space has n elements, , a probability assignment of particular interest is the case of equally likely outcomes. The probability of the elementary events is The probability of any event that consists of k outcomes, say , is Thus, if outcomes are equally likely, then the probability of an event is equal to the number of outcomes in the event divided by the total number of outcomes in the sample space. (remember the classical definition?) tch-prob

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EXAMPLE 2.6 An urn contains 10 identical balls numbered 0,1,…, 9. A random experiment involves selecting a ball from the urn and noting the number of the ball. Find the probability of the following events : A = “number of ball selected is odd,” B = “number of ball selected is a multiple of 3,” C = “number of ball selected is less than 5,” and of and tch-prob

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**If we assume that the outcomes are equally likely, then**

The sample space is , so the sets of outcomes corresponding to the above events are , , and . If we assume that the outcomes are equally likely, then . From Corollary 5. tch-prob

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**where we have used the fact that , so . From Corollary 6,**

tch-prob

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***If we count the number of heads in three tosses, then**

EXAMPLE 2.7 Suppose that a coin is tossed three times. If we observe the sequence of heads and tails, then there are eight possible outcomes If we assume that the outcomes of S3 are equiprobable, then the probability of each of the eight elementary events is 1/8. This probability assignment implies that the probability of obtaining two heads in three tosses is, by Corollary 3, Many probability models can be devised for the same sample space and events by varying the probability assignment. However, the probability assignment should be selected to reflect experimental observations to the extent possible. *If we count the number of heads in three tosses, then tch-prob

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**Continuous Sample Spaces EXAMPLE 2.11 **

Consider Experiment E12, where we picked two number x and y at random between zero and one. The sample space is then the unit square shown in Fig. 2.8(a). If we suppose that all pairs of numbers in the unit square are equally likely to be selected, then it is reasonable to use a probability assignment in which the probability of any region R inside the unit square is equal to the area of R. Find the probability of the following events: , , and Figures 2.8(b) through 2.8(c) show the regions corresponding to the events A, B, and C. Clearly each of these regions has area ½. Thus tch-prob

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**FIGURE 2.8 a Two-dimensional sample space and three events.**

(b) Event x y y x y S x (a) Sample space (c) Event x y (d) Event x y tch-prob

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**2.3 Computing probabilities using counting methods **

In many experiments with finite sample space, the outcomes can be assumed to be equiprobable. Prob Counting the number of outcomes in an event Sampling with Replacement and with Ordering choose k objects from a set A that has n distinct objects, with replacement and note the order. The experiment produces an ordered k-tuple where and The number of distinct ordered k-tuples = Ex balls, select 2 Pr[2 balls with same number] = tch-prob

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**Pr [1st number >2nd number ]= 10/20**

Sampling without Replacement and with ordering choose k objects from a set A that has n distinct objects, without replacement. number of distinct ordered k-tuples = n (n-1)…(n-k+1) Ex balls, select 2 Pr [1st number >2nd number ]= 10/20 tch-prob

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**Permutations of n Distinct Objects Consider the case when k=n **

number of permutations of n objects = n! For large n, stirling’s formula is useful Ex balls randomly placed into 12 cells, where more than 1 ball is allowed to occupy a cell. Pr [ all cells are occupied ]= ~ tch-prob

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**3. Sampling without Replacement and without ordering **

~ ” combination of size k” = each combination has k! possible orders “binomial coefficient”, read “n choose k” Ex The number of distinctive permutation of k white balls and n-k black balls From n distinct positions, pick k for white balls Partition the set of n distinctive objects into two sets tch-prob

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**Ex. 2.19 A batch of 50 items contains 10 defective items. **

Select 10 items at random, Pr [ 5 out of 10 defective ]=? Exercise prob. 47: Multinomial Ex Toss a die 12 times. Pr [ each number appears twice ]= ? {1,1,2,2,3,3, …….6,6} permutation Ex. 2.19: get 10 items in a shot (Permutation of similar items results in combinatorial (Binomial or multinomial) formula) Ex. 2.20: the die is tossed 12 times, a sequential experiment. use coin, tossed 4 times to explain tch-prob

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**4. Sampling with Replacement and without ordering**

Suppose k=5, n=4 object XX X XX The result can be summarized as XX // X / XX = k: x n-1: / = Ex Place k balls into n cells. Pick k balls from an urn of n balls. (k may be greater than n) tch-prob

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**2.4 Conditional Probability **

We are interested in knowing whether A and B are related, i.e. , if B occurs, does it alter the likelihood of A occurs. Define the conditional probability as for P[B]>0 - renormalize tch-prob

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**Example 2.21. A ball is selected from an urn containing 2 black balls (1,2)**

and 2 white balls (3,4). A: Black ball selected ½ B: even-numbered ball ½ C: number > ½ Not related Related tch-prob

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**is useful in finding prob. in sequential experiments**

Ex An urn contains 2 black, 3 white balls. Two balls are selected without replacement. Sequence of colors is noted. Sol. 1. b b two subexperiments b w w b w w Tree diagram Cannot use 2.3.3, sampling without replacement and with ordering. Why? Some balls are not distingushable. tch-prob

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**S B1 B2 Bn A Let be mutually exclusive, whose union = S ,**

partition of S. “ Theorem on total probability” Ex In prev. example (2.22) find P[W2]=Pr [second ball is white] S B1 B2 Bn A In 2.24, find out what A and Bi correspond to. tch-prob

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**Before experiment: P[ Bj ]= a priori probability **

Suppose event A occurs, what is the prob. of event Bj? Bayes’ Rule Before experiment: P[ Bj ]= a priori probability After experiment: A occurs P[ Bj/A ]= a posteriori probability tch-prob

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**Ex.2.26. Binary symmetric channel Suppose**

Ak is input symbol event Bk is output symbol event tch-prob

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**Two events A and B are independent if**

2.5 Independence of Events If knowledge of the occurrence of an event B does not alter the probability of some other event A, then A is independent of B. Two events A and B are independent if tch-prob

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**Ex. 2.28. A ball is selected from an urn of 4 balls **

{1,b},{2,b},{3,w},{4,w} A: black ball is selected B: even-numbered ball C: number >2 P[A]=P[B]=P[C]=0.5 If two events have non zero prob., and are mutually exclusive, then they cannot be independent. tch-prob

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**Three events A, B, and C are independent if**

Ex Common application of the independence concept is in making the assumption that the event of separate experiments are independent. Ex A fair coin is tossed three times. P[HHH]=1/8, …. I, 2, 3 do not necessarily imply 4 tch-prob

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**2.6. Sequential Experiments Sequence of independent experiments **

If sub experiments are independent, and event concerns outcomes of the kth subexperiment, then are independent. Bernoulli trial: Perform an experiment once, note whether an event A occurs. success, if A occurs prob= p failure, otherwise prob= 1-p Pn(k)= prob. of k successes in n independent repetitions of a Bernoulli trial = ? tch-prob

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**Ex.2.34. toss a coin 3 times. P[ H ]= p**

Binomial probability law k=0,1,…,n Bernoulli repeats n times binomial tch-prob

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**Ex.2.37. A binary channel with BER . 3 bits are transmitted. **

Binomial theorem Let a=b=1 , Let Ex A binary channel with BER bits are transmitted. majority decoding is used. What is prob. of error? tch-prob

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**Multinomial Probability Law Bj’s: partition of S; mutually exclusive**

n independent repetitions of experiment number of times Bj occurs. Ex dart areas throw dart 9 times, P[3 on each area]=? tch-prob

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**P[The initial k trials are failures]**

Geometric Probability Law Repeat independent Bernoulli trials until the occurrence of the first success. P[ more than k trials are required before a success occurs ] P[The initial k trials are failures] tch-prob

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**Urn 1 Urn 2 Trellis diagram Sequence of Dependent Experiments**

Ex A sequential experiment involves repeatedly drawing a ball from one of two urns, noting the number on the ball, and replacing the ball in its urn. Urn 0 : 1 ball #1, 2 balls #0 Urn 1 : 5 balls #1, 1 ball #0 Initially, flip a coin. Thereafter, the urn used in the subexperiment corresponds to the number selected in the previous subexperiment. Urn 1 Urn 2 Trellis diagram tch-prob

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conditional prob. Markov property Ex calculate with tch-prob

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**which four are aces and four are kings. **

Problem 95 on Page 83 Consider a well-shuffled deck of cards consisting of 52 distinct cards, of which four are aces and four are kings. a. Pr[Obtaining an ace in the first draw]= b. Draw a card from the deck and look at it. Pr[Obtaining an ace in the second draw]= If does not know the previous outcome tch-prob

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**c. Suppose we draw 7 cards from the deck. Pr[7 cards include 3 aces]=**

Pr[7 cards include 2 kings]= Pr[7 cards includes 3 aces and 2 kings]= d. Suppose the entire deck of cards is distributed equally among four players. Pr[each player gets an ace]= tch-prob

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**p =No two people in a group of n people will have a common birthday.**

Birthday problem. p =No two people in a group of n people will have a common birthday. Lotto Problem. Six numbers and one special number are picked from a pool of 42 numbers. … What is the probability of winning the grand prize? What is the probability of winning 2nd prize? What is the probability that number 37 is drawn next time? 1/(42,6) (6,5)*1/(42,6) 7/42 or 6/42 tch-prob

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