finite probability space set (sample space) function p: r + (probability distribution) p(x) = 1 x

Finite probability space

set (sample space)function P: R+ (probability distribution)

P(x) = 1x

Finite probability space


elements of are called atomic eventssubsets of are called events

probability of an event A is

P(x)xA

P(A)=

P(x) = 1x

Examples

1. Roll a (6 sided) dice. What is the probability that the number on the dice is even?

2. Flip two coins, what is the probability thatthey show the same symbol?

3. Flip five coins, what is the probability thatthey show the same symbol?

4. Mix a pack of 52 cards. What is the probability that all red cards come before all black cards?

Union bound

P(A B) P(A) + P(B)P(A1 A2 … An) P(A1) + P(A2)+…+P(An)

Union bound P(A1 A2 … An) P(A1) + P(A2)+…+P(An)

Suppose that the probability of winning ina lottery is 10-6. What is the probability thatsomebody out of 100 people wins?

Ai = i-th person wins

somebody wins = ?



Ai = i-th person wins

somebody wins = A1A2…A100



P(A1A2…A100) 100*10-6 = 10-4



P(A1A2…A100) 100*10-6 = 10-4

P(A1A2…A100) = 1–P(AC

1 AC2… AC

100) =1-P(AC

1)P(AC2)…P(AC

100)=1-(1-10-6)100 0.99*10-4

Independence Events A,B are independent if

P(A B) = P(A) * P(B)


P(A B) = P(A) * P(B)

“observing whether B happened gives no information on A”

B

A


P(A B) = P(A) * P(B)

“observing whether B happened gives no information on A”

B

AP(A|B) = P(AB)/P(B)conditional probability of A, given B


P(A B) = P(A) * P(B)

P(A|B) = P(A)

Examples

Roll two (6 sided) dice. Let S be their sum. 1) What is that probability that S=7 ? 2) What is the probability that S=7, conditioned on S being odd ? 3) Let A be the event that S is even and B the event that S is odd. Are A,B independent? 4) Let C be the event that S is divisible by 4. Are A,C independent? 5) Let D be the event that S is divisible by 3. Are A,D independent?

Examples

A

BC

Are A,B independent ?Are A,C independent ?Are B,C independent ?Is it true that P(ABC)=P(A)P(B)P(C)?

Examples

A

BC

Are A,B independent ?Are A,C independent ?Are B,C independent ?Is it true that P(ABC)=P(A)P(B)P(C)?

Events A,B,C are pairwise independent but not (fully) independent

Full independence

Events A1,…,An are (fully) independentIf for every subset S[n]:={1,2,…,n}

P ( Ai ) = P(Ai)iS iS

Testing equality of strings

Alice: A = 0001110100010101000111

Bob : B = 0001110100010101000111

slow network

QUESTION: Is A=B?

n-bits

n-bits


slow network

QUESTION: Is A=B?

Alice: A = 0001110100010101000111 Bob : B = 0001110100010101000111

n-bitsn-bits

Protocol: 1. Alice picks a random prime p n2. 2. Alice computes a:=(A mod p), and sends p and a to Bob. 3. Bob computes b:=(B mod p), and checks whether a=b.



How many bits are communicated?



What is the probabilty of failure?



What is the probabilty of failure?

BAD EVENT = p divides A-B

Testing equality of strings What is the probabilty of failure?


How many (different) primes can divide an n-bit number?

How many primes n2 are there?

Testing equality of strings What is the probabilty of failure?


How many (different) primes can divide an n-bit number?

2n M=p1p2…pk 2k k n

How many primes n2 are there?

Prime Number Theorem (m) m/ln m

number of primes m


If A=B then the algorithm always answers YES

If AB then the algorithms answers NO with probability 1- (ln n)/n

Monte Carlo algorithm with 1-sided error

Random variable


P(x) = 1x

A random variable is a function Y : RThe expected value of Y is

E[X] := P(x)* Y(x) x

Examples

Roll two dice. Let S be their sum.

If S=7 then player A gives player B $6otherwise player B gives player A $1

2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12

Examples

Roll two dice. Let S be their sum.

If S=7 then player A gives player B $6otherwise player B gives player A $1

2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12

-1 , -1,-1 ,-1, -1, 6 ,-1 ,-1 , -1 , -1 , -1

Expected income for B E[Y] = 6*(1/6)-1*(5/6)= 1/6

Y:

Linearity of expectation

E[X Y] E[X] + E[Y]E[X1 X2 … Xn] E[X1] + E[X2]+…+E[Xn]


Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get backthe card with your name – I pay you $10.

Let n be the number of people in the class.For what n is the game advantageous for me?



X1 = -9 if player 1 gets his card back 1 otherwise

E[X1] = ?



X1 = -9 if player 1 gets his card back 1 otherwise

E[X1] = -9/n + 1*(n-1)/n



X1 = -9 if player 1 gets his card back 1 otherwise X2 = -9 if player 2 gets his card back 1 otherwise

E[X1+…+Xn] = E[X1]+…+E[Xn] = n ( -9/n + 1*(n-1)/n ) = n – 10.

Expected number of coin-tosses until HEADS?


1/2 1 1/4 21/8 31/16 4….

n.2-n = 2

n=1


S

S= 1 + ½*S

S=2

Expected number of dice-throws until you get “6”

S

Expected number of dice-throws until you get “6”

S

S= 1 + (5/6)*S

S=6

Coupon collector problem

n coupons to collect

What is the expected number of cereal boxes that you need to buy?

Expected number of coin-tosses until 3 consecutive HEADS?

Markov’s inequality

A group of 10 people have average income $20,000. At most how many people in the group can have average income at least $40,000?

A group of 10 people have average income $20000. At most how many people in the group can have average income at least $100,000?

Markov’s inequality

A group of 10 people have average income $20,000. At most how many people in the group can have average income at least $40,000?

Let X be a random variable such that X 0.Then

P(X a*E[X]) 1/a

Example Alice has an algorithm A which runs inexpected running time T(n).

Bob uses Alice’s algorithm to constructhis own algorithm B. 1. Run algorithm A for 2T(n) steps. 2. If A terminates then B outputs the same, otherwise goto step 1.

What is the expected running time of B?What is the probability that A terminates after 100T(n) steps?What is the probability that B terminates after 100T(n) steps?

finite probability space set (sample space) function p: r + (probability distribution) p(x) = 1 x

Documents

pa slide

pa b c

pa n slide

union bound pa b pa

b independent

independent slide

pa bpb conditional probability

c independent