Finite probability space set function P R Px

Finite probability space set function P: R+ P(x) = 1 x (sample space) (probability distribution)

Finite probability space set function P: R+ P(x) = 1 (sample space) (probability distribution) x elements of are called atomic events subsets of are called events probability of an event A is P(A)= x A P(x)

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 that they show the same symbol? 3. Flip five coins, what is the probability that they 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(A 1 A 2 … An) P(A 1) + P(A 2)+…+P(An)

Union bound P(A 1 A 2 … An) P(A 1) + P(A 2)+…+P(An) Suppose that the probability of winning in a lottery is 10 -6. What is the probability that somebody out of 100 people wins? Ai = i-th person wins somebody wins = ?

Union bound P(A 1 A 2 … An) P(A 1) + P(A 2)+…+P(An) Suppose that the probability of winning in a lottery is 10 -6. What is the probability that somebody out of 100 people wins? Ai = i-th person wins somebody wins = A 1 A 2 … A 100

Union bound P(A 1 A 2 … An) P(A 1) + P(A 2)+…+P(An) Suppose that the probability of winning in a lottery is 10 -6. What is the probability that somebody out of 100 people wins? P(A 1 A 2 … A 100) 100*10 -6 = 10 -4

Union bound P(A 1 A 2 … An) P(A 1) + P(A 2)+…+P(An) Suppose that the probability of winning in a lottery is 10 -6. What is the probability that somebody out of 100 people wins? P(A 1 A 2 … A 100) 100*10 -6 = 10 -4 P(A 1 A 2 … A 100) = 1–P(AC 1 AC 2 … AC 100) = 1 -P(AC 1)P(AC 2)…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)

Independence Events A, B are independent if P(A B) = P(A) * P(B) “observing whether B happened gives no information on A” B A

Independence Events A, B are independent if P(A B) = P(A) * P(B) “observing whether B happened gives no information on A” B A P(A|B) = P(A B)/P(B) conditional probability of A, given B

Independence Events A, B are independent if 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 C B Are A, B independent ? Are A, C independent ? Are B, C independent ? Is it true that P(A B C)=P(A)P(B)P(C)?

Examples C Events A, B, C are pairwise independent A but not (fully) independent B Are A, B independent ? Are A, C independent ? Are B, C independent ? Is it true that P(A B C)=P(A)P(B)P(C)?

Full independence Events A 1, …, An are (fully) independent If for every subset S [n]: ={1, 2, …, n} P( Ai ) = P(Ai) i S

Testing equality of strings n-bits Alice: A = 0001110100010101000111 slow network Bob : B = 0001110100010101000111 n-bits QUESTION: Is A=B?

Testing equality of strings n-bits Alice: A = 0001110100010101000111 n-bits Bob : B = 0001110100010101000111 slow network QUESTION: Is A=B? Protocol: 1. Alice picks a random prime p n 2. 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.

Testing equality of strings How many bits are communicated? Protocol: 1. Alice picks a random prime p n 2. 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.

Testing equality of strings What is the probabilty of failure? Protocol: 1. Alice picks a random prime p n 2. 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.

Testing equality of strings What is the probabilty of failure? BAD EVENT = p divides A-B Protocol: 1. Alice picks a random prime p n 2. 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.

Testing equality of strings What is the probabilty of failure? BAD EVENT = p divides A-B How many (different) primes can divide an n-bit number? How many primes n 2 are there?

Testing equality of strings What is the probabilty of failure? BAD EVENT = p divides A-B How many (different) primes can divide an n-bit number? n 2 M=p 1 p 2…pk 2 k k n How many primes n 2 are there? Prime Number Theorem (m) m/ln m number of primes m

Testing equality of strings If A=B then the algorithm always answers YES If A B then the algorithms answers NO with probability 1 - (ln n)/n Monte Carlo algorithm with 1 -sided error

Random variable set function P: R+ P(x) = 1 (sample space) (probability distribution) x A random variable is a function Y: R The 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 $6 otherwise 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 $6 otherwise player B gives player A $1 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 Y: -1 , -1, -1, 6 , -1 , -1 Expected income for B E[Y] = 6*(1/6)-1*(5/6)= 1/6

Linearity of expectation E[X + Y] = E[X] + E[Y] E[X 1+ X 2+ … + Xn] = E[X 1] + E[X 2]+…+E[Xn]

Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the 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?

Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. X 1 = -9 if player 1 gets his card back 1 otherwise E[X 1] = ?

Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. X 1 = -9 if player 1 gets his card back 1 otherwise E[X 1] = -9/n + 1*(n-1)/n

Linearity of expectation Everybody pays me $1 and writes their name on a card. I mix the cards and give everybody one card. If you get back the card with your name – I pay you $10. X 1 = -9 if player 1 gets his card back 1 otherwise X 2 = -9 if player 2 gets his card back 1 otherwise E[X 1+…+Xn] = E[X 1]+…+E[Xn] = n ( -9/n + 1*(n-1)/n ) = n – 10.

Expected number of coin-tosses until HEADS?

Expected number of coin-tosses until HEADS? 1/2 1/4 1/8 1/16 …. 1 2 3 4 n. 2 -n = 2 n=1

Expected number of coin-tosses until HEADS? 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 in expected running time T(n). Bob uses Alice’s algorithm to construct his own algorithm B. 1. Run algorithm A for 2 T(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 100 T(n) steps? What is the probability that B terminates after 100 T(n) steps?