Discrete probability
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Discrete Probability. Hsin-Lung Wu Assistant Professor Advanced Algorithms 2008 Fall. Sample space (set) S of elementary event eg. The 36 ways of 2 dices can fall An event A is a subset of S eg. Rolling 7 with 2 dices

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Discrete Probability

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Discrete probability

Discrete Probability

Hsin-Lung Wu

Assistant Professor

Advanced Algorithms 2008 Fall


Discrete probability

  • Sample space (set) S of elementary event

    • eg. The 36 ways of 2 dices can fall

  • An eventA is a subset of S

    • eg. Rolling 7 with 2 dices

  • A probability distributionPr{} is a map from events of S to R

  • Probability Axiom:


Discrete probability

  • A random variable (r.v.) X is a function from S to R

    • The event “X = x” is defined as {sS : X(s) = x}

    • eg. Rolling 2 dices:

      • |S|=36 possible outcomes

      • Uniform distribution: Each element has the same probability

        1/|S|=1/36

      • Let X be the sum of dice

        Pr{ X = 5 } = 4/36, {(1, 4), (2, 3), (3, 2), (4, 1)}

  • Expected value:

  • Linearity:

    • X1: number on dice 1

    • X2: number on dice 2

    • X=X1+X2, E[X1]=E[X2]=1/6(1+2+3+4+5+6)=21/6


Independence

Independence

  • Two random variables X and Y are independent if


Discrete probability

  • Indicator random variables

    • Given a sample space S and an event A, the indicator random variable I{A} associated with event A is defined as:


Discrete probability

  • E.g.: Consider flipping a fair coin:

    • Sample space S = { H,T }

    • Define random variableY with Pr{ Y=H } = Pr{ Y=T }=1/2

    • We can define an indicator r.v.XH associated with the coin coming up heads, i.e.Y=H


Discrete probability

  • The birthday paradox:

    • How many people must there be in a room before there is a 50% chance that two of them born on the same day of the year?

  • (1)

    • Suppose there are k people and there are n days in a year,bi : i-th person’s birthday, i =1,…,k

    • Pr{bi=r}=1/n, for i =1,…,k and r=1,2,…,n

    • Pr{bi=r, bj=r}=Pr{bi=r}.Pr{bj=r} = 1/n2


Discrete probability

  • Define event Ai : Person i’s birthday is different from person j’s for j < i

  • Pr{Bk}= Pr{Bk-1∩Ak}= Pr{Bk-1}Pr{Ak|Bk-1}where Pr{B1}= Pr{A1}=1


Discrete probability

  • (2) Analysis using indicator random variables

    • For each pair (i, j) of the k people in the room, define the indicator r.v.Xij, for 1≤ i < j ≤ k, by


Discrete probability

  • When k(k-1)≥ 2n, the expected number of pairs of people with the same birthday is at least 1


Discrete probability

  • Balls and bins problem:

    • Randomly toss identical balls into b bins, numbered 1,2,…,b. The probability that a tossed ball lands in any given bin is 1/b

    • (a) How many balls fall in a given bin?

      • If n balls are tossed, the expected number of balls that fall in the given bin is n/b

    • (b) How many balls must one toss, on the average, until a given bin contains a ball?

      • By geometric distribution with probability 1/b


Discrete probability

  • (c) (Coupon collector’s problem)How many balls must one toss until every bin contains at least one ball?

    • Want to know the expected number n of tosses required to get b hits. The ith stage consists of the tosses after the (i-1)st hit until the ith hit.

    • For each toss during the ith stage, there are i-1 bins that contain balls and b-i+1 empty bins

    • Thus, for each toss in the ith stage, the probability of obtaining a hit is (b-i+1)/b

    • Let ni be the number of tosses in the ith stage. Thus the number of tosses required to get b hits is n=∑bi=1 ni

    • Each ni has a geometric distribution with probability of success (b-i+1)/b→ E[ni]=b/b-i+1


Streaks

Streaks

  • Flip a fair coin n times, what is the longest streak of consecutive heads?Ans:θ(lg n)

  • Let Aik be the event that a streak of heads of length at least k begins with the ith coin flip

  • For j=0,1,2,…,n, let Lj be the event that the longest streak of heads has Length exactly j, and let L be the length of the longest streak.


Discrete probability

  • We look for streaks of length s by partitioning the n flips into approximately n/s groups of s flips each.


Discrete probability

s

s

s

n

  • The probability that a streak of heads of length does not begin in position i is


Discrete probability

  • Using indicator r.v. :


Discrete probability

  • If c is large, the expected number of streaks of length clgn is very small.

  • Therefore, one streak of such a length is very likely to occur.


The on line hiring problem

The on-line hiring problem:

  • To hire an assistant, an employment agency sends one candidate each day. After interviewing that person you decide to either hire that person or not. The process stops when a person is hired.

  • What is the trade-off between minimizing the amount of interviewing and maximizing the quality of the candidate hired?


The on line hiring problem1

The on-line hiring problem:

Pi

Pk-1

Pk-1

<?

P2

P1

Pk

….


Discrete probability

What is the best k?


Discrete probability

  • Let M(j) = max 1ij{score(i)}.

  • Let S be the event that the best-qualified applicant is chosen.

  • Let Si be the event the best-qualified applicant chosen is the i-th one interviewed.

  • Si are disjoint and we have Pr{S}=  ni=1Pr{Si}.

  • If the best-qualified applicant is one of the first k, we have that Pr{Si}=0 and thus

  • Pr{S}=  ni=k+1Pr{Si}.


Discrete probability

  • Let Bi be the event that the best-qualified applicant must be in position i.

  • Let Oi denote the event that none of the applicants in position k+1 through i-1 are chosen

  • If Si happens, then Bi and Oi must both happen.

  • Bi and Oi are independent! Why?

  • Pr{Si} = Pr{Bi  Oi} = Pr{Bi} Pr{Oi}.

  • Clearly, Pr{Bi} = 1/n.

  • Pr{Oi} = k/(i-1). Why???

  • Thus Pr{Si} = k/(n(i-1)).


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