CMU 15-251 Probability I Teachers: Anil Ada Ariel Procaccia (this - - PowerPoint PPT Presentation

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CMU 15-251 Probability I Teachers: Anil Ada Ariel Procaccia (this - - PowerPoint PPT Presentation

Sep 25, 2022 113 likes β€’400 views

CMU 15-251 Probability I Teachers: Anil Ada Ariel Procaccia (this time) Course overview 2 Gambling 101 5 7 8 3 5 5 5 1 7 1 8 2 8 3 3 3 5 7 2 9 5 7 2 9 3 Gambling 101 4 5 1 = 0.518

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SLIDE 1

CMU 15-251

Probability I

Teachers: Anil Ada Ariel Procaccia (this time)

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SLIDE 2

Course overview

2

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SLIDE 3

Gambling 101

  • 3

7 7 1 7 7 1 5 5 5 5 5 5 3 3 3 9 9 3 8 2 8 2 2 8

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SLIDE 4

Gambling 101

  • 1 βˆ’

5 6 4

= 0.518

4

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SLIDE 5

Gambling 101

  • 1 βˆ’

35 36 24

= 0.491

  • 5
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SLIDE 6

Gambling 101

6

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SLIDE 7

Pennies and gold

  • 1.

1/6

2.

1/3

3.

2/3

4.

1

7

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SLIDE 8

Language of probability

8

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SLIDE 9

Language of probability

  • 𝑇
  • π‘ž

π‘¦βˆˆπ‘‡

π‘ž 𝑦 = 1

9

0.1 0.5 𝑇 0.2 0.2

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SLIDE 10

Language of probability

  • 𝐹 βŠ† 𝑇
  • Pr 𝐹 = π‘¦βˆˆπΉ π‘ž(𝑦)
  • 𝑦 ∈ 𝑇

Pr 𝐹 =

π‘¦βˆˆπΉ

π‘ž 𝑦 = |𝐹| |𝑇|

10

0.1 0.2 0.5 0.2 𝑇 Pr 𝐹 = 0.6

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SLIDE 11

Language of probability

  • 𝑇 =

{ 1,1 , 1,2 , 1,3 , 1,4 , 1,5 , 1,6 , 2,1 , 2,2 , 2,3 , 2,4 , 2,5 , 2,6 , 3,1 , 3,2 , 3,3 , 3,4 , 3,5 , 3,6 , 4,1 , 4,2 , 4,3 , 4,4 , 4,5 , 4,6 , 5,1 , 5,2 , 5,3 , 5,4 , 5,5 , 5,6 , 6,1 , 6,2 , 6,3 , 6,4 , 6,5 , (6,6)}

  • 1.

1/9

2.

2/9

3.

3/9

4.

4/9

11

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SLIDE 12

Conditional probability

  • 𝐡

𝐢 Pr 𝐡 𝐢 = Pr[𝐡∩𝐢]

Pr[𝐢]

  • 𝐡 ∩ 𝐢

𝐢

12

0.1 0.2 0.5 0.2 𝑇 𝐡 𝐢 𝐡 ∩ 𝐢

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SLIDE 13

Pennies and gold, revisited

  • 𝐻𝑗

𝑗 ∈ {1,2}

  • Pr 𝐻1 =

1 2 , Pr 𝐻1 ∩ 𝐻2 = 1 3

  • Pr 𝐻2 𝐻1 =

1/3 1/2 = 2 3

13

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SLIDE 14

Conditional probability

  • Pr 𝐡 ∩ 𝐢 = Pr 𝐢 Γ— Pr[𝐡|𝐢]
  • 𝐡

𝐢 𝐢 𝐡 𝐢

  • Applying iteratively:

Pr 𝐡1 ∩ β‹― ∩ π΅π‘œ = Pr 𝐡1 Γ— Pr 𝐡2 𝐡1 Γ— β‹― Pr[π΅π‘œ|𝐡1, β‹― , π΅π‘œβˆ’1]

14

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SLIDE 15

bayes’ rule

  • Pr B Γ— Pr A B = Pr 𝐡 ∩ 𝐢 = Pr 𝐡 Γ— Pr[𝐢|𝐡]

15

Pr 𝐡 𝐢 = Pr 𝐡 Pr[𝐢|𝐡] Pr[𝐢]

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SLIDE 16

Monty Hall problem

  • 16
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SLIDE 17

Monty Hall problem

17

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SLIDE 18

Monty Hall problem

  • Pr 𝑄3 𝑃2 =

Pr[𝑄3] Pr[𝑃2|𝑄3] Pr[𝑃2]

  • Pr[𝑄3] =

1 3 , Pr 𝑃2 𝑄3 = 1,

Pr[𝑃2] = 1/2

  • Pr 𝑄3 𝑃2 = 2/3
  • 1.

3/15

2.

4/15

3.

5/15

4.

6/15

18

Pr 𝐡 𝐢 = Pr 𝐡 Pr[𝐢|𝐡] Pr[𝐢]

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SLIDE 19

Independence

  • 𝐡

𝐢 Pr 𝐡 𝐢 = Pr[𝐡]

  • 19
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SLIDE 20

The birthday paradox

  • 𝑛
  • 𝑇 = 1, … , 365 𝑛, s

𝑦 = (𝑦1, … 𝑦𝑛)

  • 𝐹 =

𝑦 ∈ 𝑇 βˆƒπ‘—, π‘˜, s.t. 𝑦𝑗 = π‘¦π‘˜}

20

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SLIDE 21

The birthday paradox

  • 𝐹
  • 𝐹
  • 𝐡𝑗

𝑗 1, … , 𝑗 βˆ’ 1

  • 𝐹 = 𝐡1 ∩ β‹― ∩ π΅π‘œ
  • Pr

𝐹 = Pr 𝐡1 Γ— Pr 𝐡2 𝐡1 Γ— β‹― Pr[π΅π‘œ|𝐡1, β‹― , π΅π‘œβˆ’1]

21

Pr 𝐡𝑗 𝐡1, … , π΅π‘—βˆ’1 ?

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SLIDE 22

The birthday paradox

  • 𝐡1 ∩ β‹― ∩ π΅π‘—βˆ’1

𝑗 βˆ’ 1

  • 𝑗 βˆ’ 1

𝑗

  • Pr 𝐡𝑗 𝐡1, … , π΅π‘—βˆ’1 = 365βˆ’(π‘—βˆ’1)

365

= 1 βˆ’ π‘—βˆ’1

365

  • Pr

𝐹 = 1 Γ— 1 βˆ’

1 365 Γ— β‹― Γ— 1 βˆ’ π‘›βˆ’1 365

  • Pr 𝐹 = 1 βˆ’ Pr[

𝐹]

22

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SLIDE 23

The birthday paradox

23

Pr 𝐹 : Pr 𝐹 :

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SLIDE 24

The birthday paradox

  • 1.

1/2

2.

0.75

3.

0.99999999999997

4.

1

24

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SLIDE 25

Birthday attack *

  • 𝑇

𝑙 𝑔(𝑇)

  • 𝑇1, 𝑇2

𝑔 𝑇1 = 𝑔 𝑇2

  • 𝑛
  • 𝑔(𝑛)
  • 𝑛

𝑛′ 𝑔 𝑛 = 𝑔(𝑛′)

25

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SLIDE 26

Birthday attack *

  • 1, … , 2160
  • 2160 = 280

26

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SLIDE 27

Birthday attack *

  • 263
  • 27
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SLIDE 28

Summary

  • 28