1/4 2/3 B G 1/4 1/2 1 randomwalk.1 randomwalk.2 - - PowerPoint PPT Presentation

SLIDE 1

random­walk.1

Random Walks

Mathematics for Computer Science

MIT 6.042J/18.062J

Albert R Meyer, May 13, 2015 random­walk.2

Graph With Probable Transitions

Outgoing­edge probabilities sum to 1

1/4 1/4 2/3 1/3 1 1/2

B O G

Albert R Meyer, May 13, 2015 random­walk.3

Example: Gambler’s Ruin

View as random walk on a line. p ::=Pr[win a bet]

$0 n­1 n n+1 T k p k+1 k k­1 q T­1 $1

q ::= 1­p = Pr[lose a bet] What is Pr[reach T before 0]?

Albert R Meyer, May 13, 2015

Applications of Random Walk

• Physics — Brownian motion
• Finance — stocks, options
• Algorithms — web search,

clustering

random­walk.4 Albert R Meyer, May 13, 2015

1

SLIDE 2

T T T T T

random­walk.5

Questions

• Pr[reach O in 7 steps| start at B]
• Average # steps from B to O
• Pr[reach G before O | start at B]

1/4 1/4 2/3 1/3 1 1/2

B O G

Albert R Meyer, May 13, 2015 random­walk.6

Example: Toss HTH before TTH

Albert R Meyer, May 13, 2015

­­

­­ ­H

H ­­

­H ­T

random­walk.7

Example: Toss HTH before TTH

Albert R Meyer, May 13, 2015

­­

­­ ­H

H

­T

T

½ ½

­­ HH

H

HT

T

½ ½

random­walk.8

Example: Toss HTH before TTH

Albert R Meyer, May 13, 2015

­­

­­ ­H

H

­T

T

½ ½

­­ HH

H

­H HH

2

­T

T

½ ½

Pr[win] = Pr[win| ] = ½Pr[win| ] + ½Pr[win| ]

HT

T

½ ½

Pr[win| ] = ½Pr[win| ]

SLIDE 3

T T T T T T T T T T

random­walk.9

Example: Toss HTH before TTH

Albert R Meyer, May 13, 2015

­­

­­ ­H

H

­T

T

½ ½

­­ HH

H

­H HT HH

random­walk.10

Example: Toss HTH before TTH

Albert R Meyer, May 13, 2015

­­

­­ ­H

H

­T

T

½ ½

­­ HH

H

HT

T

½ ½

­­ TH

H

TT

T

random­walk.11

Example: Toss HTH before TTH

Albert R Meyer, May 13, 2015

­­

­H

H

­T

T

½ ½

­­ HH

H

HT

T

½ ½

­­ TH

H

­T TT TH

random­walk.12

Example: Toss HTH before TTH

Albert R Meyer, May 13, 2015

­­

­­ ­H

H

­T

T

½ ½

­­ HH

H

HT

T

½ ½

­­ TH

H

TT

T H T

3

HT

T

½ ½

Pr[win| ] = ½Pr[win| ] + ½Pr[win| ]

TT

T

Pr[win| ] = ½Pr[win| ] + ½Pr[win| ]

SLIDE 4

T T T T T T

random­walk.13

Example: Toss HTH before TTH

Albert R Meyer, May 13, 2015

­­

­­ ­H

H

­T

T

½ ½

­­ HH

H

HT

T

½ ½

­­ TH

H

TT

T H

HH HH HT

random­walk.14

Example: Toss HTH before TTH

Albert R Meyer, May 13, 2015

­­

­­ ­H

H

­T

T

½ ½

­­ HH

H

HT

T

½ ½

­­ TH

H

TT

T H T T

win

T H H T H

lose

T H H T

Pr[win| ] = 0 Pr[win| ] = 1

win

Now solve system of linear equations for Pr[win]

lose

random­walk.15

Questions

• Pr[reach O in 7 steps| start at B]
• Average # steps from B to O
• Pr[reach G before O | start at B]

1/4 1/4 2/3 1/3 1 1/2

B O G

Albert R Meyer, May 13, 2015

Just solve systems of linear equations

4

T

Pr[win| ] = ½Pr[win| ] + ½Pr[win| ]

SLIDE 5

MIT OpenCourseWare https://ocw.mit.edu

6.042J / 18.062J Mathematics for Computer Science

Spring 2015 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms.