Chapter gambling A simple I : game Newstead Define - - PowerPoint PPT Presentation

Chapter I

:

A

simple

gambling

game

Newstead

① Define stochastic

process

② Give

an

example

based

• n

gambling with

coin flips

③ Analyze

"

long

• term

"

behavior

• f this

example

Deth

( stochastic

Process)

A stochastic

process

is

a

indexed

collection

• f

random variables

{ Xi lien

• n (d. F. IP)

(we'll

focus

• n the

case

where I

• IN
• r I
• Hulot

,

in

which

case

we'll

view

Xn

as

representing The

state

at

some

system

at time

n

" Discrete

time

evolution stochastic

processes

")

Note :

in general ,

don't expect { Xn3neµ to be

in dcp

Exe ( counting successful

coin flips )

let

Bn

be given

by

Bn ( w)

is

the

nth

binary position

• f

w .

let

Xin

=

B, t

• - t Bn

.

Interpretation

i

① sum of fist

n

big positions of w

② total

number of

"heads

" after

n flip,

we'll

tweak

this

by

making

it

into

a simple

gambling

game

.

Exe

( Betting

• n

coin flips)

let

X.

= Sla

let

X , :{$$aH

if

B ,

• I
• a - I

if

B , =D

let

Xz

= {

X , tl

if

132=1

* ,

• I

if

132=0

let

Xs

. { YET

,

it

134

if

B>

⇐ 0

In

general

:

Xu

=

at

2113

, t

• - 7- Bn)
• n

= at (2B ,

• 1) + (2132
• 1) t
• - -t ( 2B.
• i)

Here

Xa

is

the

amount of

money

after

n

coin flips

Exe

( Belting

an biased

coin flips)

ht

{ Cn)

be

iid

random

vanning

with

BCG : ' )

• p

& Pkn:o)

• q

where

ptg

• l .

let

• X. =$a

let

X .

• Haat

,

it

9 "

C,

• O

let

Xz

= {

X , tl

if

• C. z
• I

* ,

• I

if 62=0

In

general

:

Xu

=

at

21Gt

• - tch )
• n

= at (24-1)+(24-1) t

• - -t ( 24
• i)

Here

Xa

is

The amount of

money

after

n

coin flips

Then

( Coia

flip gambling

density )

IP ( Xu

=

at K)

= ( ing ) pntklz qn

• 142

for

• ne KE n

and

ntk

is

even .

PI If

in

n

flips

we

have l

successes

,then

we

have

n

• l

failures,

and

so

we

have

a * l

• ( n
• l)
• at 2e
• n

dollars in

my

pocket

Hence

Xu

• atk
• nly if

ntk

is

even

To

have

Xu - atk ,

we

need

ntzk

successes

and

Ig

failure,

distributed among

n

coin flips

How

my ways

can I

place thou

htt

successes

amongst

my

n

coin

tosses ?

we

have to

choose

NII

positions

them The

n

possible positions , and

there

are (Itis) ways to

do this . Each

such

caaligvmtion

• ccurs

with probability

pntkqn

'T

( from

independence of

{ Cn 3) ④

This

is

good , but

it's

not

complete

misses

① what

is

"

long term

behavior

"

• f

Xu ?

SUN

says

IP (

"I

Xn

• a)
• I
• when

p

• Ya

① this

to

recognize the

"real world

restrictions

"

implicit

in

this

being

a gambling problem

In particular

, if

Xa

• O

, Then

we

have

to stop

playing

Defy

( hitting times)

Suppose

as c

To

• int l

as o

: * n

• o}

To

• int f

n 30 : Xn

• c }

These

are

the

" first

hitting times

" fer

O

and c.

Big question

:

what

is

RC To

a T . ) ?

How

do

we

approach this ?

Recursion

The highlight

salient

parameters

, lets

write

so, p (a)

⇐

IP ( To

< To )

We

get

so

,pla)

• IP( To

< To )

=

IP( C ,

• l

, To < To ) t IP ( CEO , To - To )

=

p

so

,platt) t q sup la

• t)

This

gives

a

recurrence

relation that

we

can

solve

for

so , p

la)

( To be completed in homework)

THI Gambler's Ruin

• Success)

so , plat

• IP ( Tc at.)

= {Y%

it

p

• the

%

it

f-

'k

Nate : if

p

• 42

and it

as

:c ,then

you

have

a

good

chance

to

win

before

going

broke .

if

path

( even by

a little ) ,you

can

be

in

trouble

Ex

p

• 0.49

a

c

lP( To - To )=

97788%

iii.of:÷f÷÷

:

• '74

Toooo

100000

2x to

Thy ( Gambler's

Ruin

• Failure)
• MT. - Ta)
• f

peek

p

• '12

C

PI

Imagine

you

have $a

at

five

O

and

sees

is

probability

p , and th

bank

has Isc

• Sla

at

time and

success probability

A

• p
• q

If

T ca To

for

you

,then

To a Tc

for

bank .

(& vice

versa)

This

means

( T. < To )

=

So,q( c-a)

• d

bank has c-a

dollars

,

"success g

and you going

broke

mean,

bank gets all

a

dollars

Bh

what

happens if

you

gamble forever , but

have to

step if

you

ever

go bake ?

E.g. , what

is

IT ( To co) ?

them (you

almost always

lose)

IP ( T. so )

= {

1

if

pe

'k

( Yp)

"

if

p

• Yz

tf

let

Hc

a- { To - To)

I HIT l T

• ca}

Now

use

continuity

• f probability ,

and

• ur

formula

for failure

version

at

gambler's ruin

• ppg