Tutorial 2 Solution for homework 1 - National Tsing Hua ...
Transcript of Tutorial 2 Solution for homework 1 - National Tsing Hua ...
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Randomized algorithm
Tutorial 2Solution for homework 1
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Outline
Solutions for homework 1Negative binomial random variablePaintball gameRope puzzle
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Solutions for homework 1
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Homework 1-1
:a
:b
:c
:dBox 1 Box 2
step 1
babB
baaW
+=
+=
)Pr(
)Pr(
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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)
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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
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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
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Homework 1-2
22
22
31
1-ba1-a
1-ba
1-a1-ba
1-a
1-ba1-a
⎟⎠⎞
⎜⎝⎛
+<<⎟
⎠⎞
⎜⎝⎛
+⇒
⎟⎠⎞
⎜⎝⎛
+<
+×
+<⎟
⎠⎞
⎜⎝⎛
+⇒
+<
+
baa
baa
baa
baa
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Homework 1-2
2)13(1
1313
31)13(
1)1(3
)1()1(3 22
ba
ba
ba
baa
baa
++<⇒
−−+
<⇒
+−<−⇒
−+<−⇒
−+<−
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Homework 1-2
ab
abab
aba
aba
<+
⇒
<−
⇒
−<⇒
<+⇒
<+
2)13(
13
)13(
3
3)( 22
2)13(1
2)13( bab +
+<<+
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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
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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
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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
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Homework 1-5
i-th pair
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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
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Homework 1-6
pqqppq
pqqppq
qqpp
kYkYkXYX
k
kk
kkk
−+=
+−−=
−−=
====
=
∑
∑
∑
−
−−
1
11
)1(
)1()1(
)Pr()|Pr()Pr(
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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
−−−−+=
−+=−−−
=
=
=
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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]
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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(
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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).
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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
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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
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Homework 1-7
e
neen
neennn
nmnm
E
e
ee
ee
1
)(log
))/(loglog(
)loglog()Pr(
=
=
−=
−≥
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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
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Negative binomial random variable
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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(
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Negative binomial random variable
Let z=y-r
kr
kr
ppkkr
ppr
krrky
kz
)1(1
)1(1
1)Pr(
)Pr(
−⎟⎟⎠
⎞⎜⎜⎝
⎛ −+=
−⎟⎟⎠
⎞⎜⎜⎝
⎛−−+
=
+===
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Negative binomial random variable
Let r=1
Isn’t it familiar?
k
kr
pp
ppkkrkz
)1(
)1(1
)Pr(
−=
−⎟⎟⎠
⎞⎜⎜⎝
⎛ −+=
=
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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.
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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
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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
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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
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Paintball game
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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.
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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.
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Paintball game
What is your best strategy of first shoot?Please calculate each player’s probability of winning.
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Rope puzzle
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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.
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Rope puzzle
There can be three cases1.
2.
3.
Which case has higher probability?