Finite probability space set (sample space) function P: R + (probability distribution) P(x) = 1 x
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Transcript of 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)
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(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 = ?
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 = A1A2…A100
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?
P(A1A2…A100) 100*10-6 = 10-4
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?
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)
“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
AP(A|B) = P(AB)/P(B)conditional probability of A, given B
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
Testing equality of strings
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.
Testing equality of strings
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?
Testing equality of strings
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.
What is the probabilty of failure?
Testing equality of strings
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.
What is the probabilty of failure?
BAD EVENT = p divides 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 n2 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?
2n M=p1p2…pk 2k k n
How many primes n2 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 AB then the algorithms answers NO with probability 1- (ln n)/n
Monte Carlo algorithm with 1-sided error
Random variable
set (sample space)function P: R+ (probability distribution)
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
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?
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 backthe card with your name – I pay you $10.
X1 = -9 if player 1 gets his card back 1 otherwise
E[X1] = ?
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 backthe card with your name – I pay you $10.
X1 = -9 if player 1 gets his card back 1 otherwise
E[X1] = -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 backthe card with your name – I pay you $10.
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.
Coupon collector problem
n coupons to collect
What is the expected number of cereal boxes that you need to buy?
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?