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?
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