Randomized algorithm

Tutorial 2Solution for homework 1

Outline

Solutions for homework 1Negative binomial random variablePaintball gameRope puzzle

Solutions for homework 1

Homework 1-1

：a

：b

：c

：dBox 1 Box 2

step 1

babB

baaW

+=

+=

)Pr(

)Pr(

Homework 1-1

Draw a white ball from Box 2 : event APr(A) = Pr(A∩ (W∪B))= Pr((A∩ W)∪(A∩ B))= Pr(A∩ W) + Pr(B∩ W) = Pr(A|W)Pr(W) + Pr(A|B)Pr(B)

Homework 1-1

It is easy to see that

1)db)(c(aabcac

B)Pr(B)|Pr(A W)Pr(W)|Pr(A Pr(A)

1dccB)|Pr(A

11W)|Pr(A

+++++

=

+=

++=

+++

=dc

c

Homework 1-2

Wi = pick a white ball at the i-th time

31

11

PrPrPr

112

21

=

+×

−+−

=

=∩

baa

baa

)(W)|W(W)W(W

Homework 1-2

22

22

31

1-ba1-a

1-ba

1-a1-ba

1-a

1-ba1-a

⎟⎠⎞

⎜⎝⎛

+<<⎟

⎠⎞

⎜⎝⎛

+⇒

⎟⎠⎞

⎜⎝⎛

+<

+×

+<⎟

⎠⎞

⎜⎝⎛

+⇒

+<

+

baa

baa

baa

baa

Homework 1-2

2)13(1

1313

31)13(

1)1(3

)1()1(3 22

ba

ba

ba

baa

baa

++<⇒

−−+

<⇒

+−<−⇒

−+<−⇒

−+<−

Homework 1-2

ab

abab

aba

aba

<+

⇒

<−

⇒

−<⇒

<+⇒

<+

2)13(

13

)13(

3

3)( 22

2)13(1

2)13( bab +

+<<+

Homework 1-2

1 b maps 1 a onlyFor b=1, 1.36<a<2.36 means a=2

Pr(W1∩W2) = 2/3 * 1/2= 1/3

Then we may tryb=2 →a=3 →oddb=4 →a=6 →even

Homework 1-3

By the hintX = sum is even.Y = the number of first die is evenZ = the number of second die is oddPr(X∩Y) = ¼ = ½ * ½Pr(X∩Z) = ¼ = ½ * ½Pr(Y∩Z) = ¼ = ½ * ½Pr(X∩Y∩Z) =0

Homework 1-4

Let C1, C2, …, Ck be all possible min-cut sets.

2)1(

1)1(

2

1) returns algorithmPr(

)1(2) returns algorithmPr(

1

−≤⇒

≤−

⇒

≤⇒

−=

∑=

nnk

nnk

C

nnC

k

ii

i

Homework 1-5

i-th pair

Homework 1-5

Indicator variablePr(Xi) = 1 if the i-th pair are couple.Pr(Xi) = 0 Otherwise

Linearity of expectationX= Σi=1 to 20Xi

E[X] = Σi=1 to 20E[Xi] = Σi=1 to 201/20 = 1

Homework 1-6

pqqppq

pqqppq

qqpp

kYkYkXYX

k

kk

kkk

−+=

+−−=

−−=

====

=

∑

∑

∑

−

−−

1

11

)1(

)1()1(

)Pr()|Pr()Pr(

Homework 1-6

)1)(1(1111

)],[min(E][E][E)],[max(E)1)(1(1

1)],[min(E

1][E

1][E

qpqp

YXYXYXqp

YX

qY

pX

−−−−+=

−+=−−−

=

=

=

Homework 1-7

To choose the i-th candidate, we needi>mi-th candidate is the best [Event B]The best of the first i-1 candidates should be in 1 to m. [Event Y]

Homework 1-7

Therefore, the probability that i is the best candidate is

The probability of selecting the best one is

11)|Pr()Pr()Pr(

−×==

im

nBYBE iiii

∑∑+=+= −

==n

mi

n

mii in

mEE11 1

1)Pr()Pr(

Homework 1-7

Consider the curve f(x)=1/x. The area under the curve from x = j-1 to x = j is less than 1/(j-1), but the area from x = j-2 to x = j-1 is larger than 1/(j-1).

Homework 1-7

mndxxfj ee

n

m

n

mjloglog)(

11

1−=≥

− ∫∑+=

)1(log)1(log)(1

1 1

11

−−−=≤− ∫∑

−

−+=

mndxxfj ee

n

m

n

mj

Homework 1-7

mnmg

nnmnmg

mnnmmg

ee

ee

1)(''

1loglog)('

)log(log)(

−=

−−

=

−=

→g’(m)=0 when m=n/e

→g’’(m)<0 , g(m) is max when m=n/e

Homework 1-7

e

neen

neennn

nmnm

E

e

ee

ee

1

)(log

))/(loglog(

)loglog()Pr(

=

=

−=

−≥

Homework 1-8 (bonus)

S={±1, ±2, ±3, ±4, ±5, ±6}X={1,-2,3,-4,5,-6}Y={-1,2,-3,4,-5,6}E[X]=-0.5E[Y]=0.5E[Z] = -3.5

Negative binomial random variable

Negative binomial random variable

Definitiony is said to be negative binomial random variable with parameter r and p if

For example, we want r heads when flipping a coin.

rkr pprk

ky −−⎟⎟⎠

⎞⎜⎜⎝

⎛−−

== )1(11

)Pr(

Negative binomial random variable

Let z=y-r

kr

kr

ppkkr

ppr

krrky

kz

)1(1

)1(1

1)Pr(

)Pr(

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−+

=

+===

Negative binomial random variable

Let r=1

Isn’t it familiar?

k

kr

pp

ppkkrkz

)1(

)1(1

)Pr(

−=

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

=

Negative binomial random variable

Suppose the rule of meichu game has been changed. The one who win three games first would be the champion. Let p be the probability for NTHU to win each game, 0<p<1.

Negative binomial random variable

Let the event

We know

∑==

==5

3

5

3

)()Pr() winsNTHUPr(k

kk

k APAU

5,4,3 }gameth - on the winsNTHU{ == kkAk

Negative binomial random variable

where

Hence33

5

3)1(

21

) winsNTHUPr( −

=

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −=∑ k

k

ppk

33 )1(2

1)gameth on success rd3()Pr( −−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −== k

k ppk

kPA

Negative binomial random variable

Given that NTHU has already won the first game. What is the probability of NTHU being the champion?

1611

163

82

41

21

21

11 224

2=⎟

⎠⎞

⎜⎝⎛ ++=⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ − −

=∑

k

k

k

Paintball game

Paintball game

You James Bond, and Robin Hood decide to play a paintball game. The rules are as follows

Everyone attack another in an order.Anyone who get “hit” should quit.The survival is the winner.You can choose shoot into air.

Paintball game

James Bond is good at using gun, he can hit with probability 100%.Robin Hood is a bowman, he can hit with probability 60%.You are an ordinary person, can only hit with probability 30%. The order of shot is you, Robin Hood then James Bonds.

Paintball game

What is your best strategy of first shoot?Please calculate each player’s probability of winning.

Rope puzzle

Rope puzzle

We have three ropes with equal length (Obviously, there are 6 endpoint).Now we randomly choose 2 endpoints and tied them together. Repeat it until every endpoint are tied.

Rope puzzle

There can be three cases1.

2.

3.

Which case has higher probability?