Chapter gambling A simple I : game
Newstead ① Define stochastic process based example ② Give an gambling with on coin flips ③ Analyze behavior " " long - term example of this
Process ) Deth ( stochastic collection indexed is A stochastic process a on ( d. F. IP ) { Xi lien variables random of . - Hulot or I - IN on the where I - ( we 'll - focus case , representing The we 'll Xn as view which case in at time system n at state . some " ) stochastic processes evolution " Discrete time { Xn3ne µ to be in dcp don't expect in general , . Note :
coin flips ) Exe ( counting successful Bn ( w ) be given is by : lo , is → { Oil ) Bn let of position binary w . nth the - - t Bn B , t Xin let = . big positions of w ① sum of fist Interpretation n " after i n flip , " heads number of ② total a simple into it making by this we'll tweak gambling game .
coin flips ) ( Betting Exe on - I if B , X , :{ $$aH - = Sla let X. let B , =D if - a - I . = { 132=1 if X , tl Xz let if 132=0 - I * , 134 it . { YET Xs let if B > ⇐ 0 , - - 7- Bn ) 2113 - n , t at Xu general = In : - 1) t - i ) - - - t ( 2B . - 1) + ( 2132 = at ( 2B , coin flips after amount of Xa the Here is money n
coin flips ) ( Belting biased Exe an BCG : ' ) & Pkn :o) - q - p with - vanning random iid - { Cn ) be ht - l . ptg where - - Haat it 9 " X. =$a X . let let - - O C , . , - = { - I C. z if X , tl Xz let if 62=0 - I * , - - tch ) 21Gt - n at Xu general = In : - i ) = at (24-1)+(24-1) t - - - t ( 24 coin flips after amount of Xa The Here is money n
density ) flip gambling ( Coia Then = ( ing ) pntklz qn -142 at K ) IP ( Xu = ntk even . and is for - ne KE n , then have l successes we flips PI If we in n have and we so failures , - l have n - at 2e - ( n - l ) - n a * l - only if - atk Hence Xu pocket dollars in - my . is even ntk ,
need and Xu - atk , ntzk have successes we To coin flips distributed Ig failure , n among . htt place thou can I ways How my tosses ? coin n amongst my successes them The positions NII choose to have are ( Itis ) ways to we positions , and there possible n with caaligvmtion do this . Each such occurs { Cn 3) independence of 'T pntkqn ( from probability ④
complete not good , but it's . It misses This is of Xu ? behavior " long term ① what " is - a ) IP ( - I " I Xn - says SUN - - Ya when - p restrictions " " real world recognize the ① this to a gambling problem being . this implicit in have - O , if , Then we Xa In particular - playing to stop .
⇐ ( hitting times ) Defy - o } - int l : * n - as o . let To - Suppose as c - c } - int f : Xn n 30 - To - " fer " first and c . hitting times O the These are a T . ) ? RC To Big question what is : approach this ? Recursion do . How we , lets write parameters salient The highlight < To ) so , p (a) IP ( To .
< To ) IP ( To get , pla ) We so - - , To < To ) t IP ( CEO , To - To ) IP ( C , - l - = - t ) , platt ) t q sup la = so p that relation can we gives recurrence This a ( To be completed in homework ) la ) for so , p solve . Success ) THI Gambler 's Ruin = { Y% - it the p - IP ( Tc at . ) so , plat - - it ' k % f-
have and it - 42 Nate : if , then a you as :c p - broke . before going to good chance win trouble a little ) , you be in by can . ( even if path lP( To - To )= iii. of :÷f÷÷ - 0.49 Ex c a - p 97788% : - ' 74 2x to 100000 Toooo
- f Failure ) Ruin Thy ( Gambler 's - peek MT . - Ta ) - ' 12 p - - C and sees O five at $a have PI Imagine you - Sla at Isc bank p , and th has probability - q is - p A success probability - and . 0 time for bank . To a Tc for , then If ca To you T versa ) ( & vice
( T . < To ) So ,q( c- a ) This means = - d bank has c- a " " success g dollars , and you going mean , broke bank gets all a dollars Bh
gamble forever , but have to happens if what you bake ? step if go ever you IT ( To co ) ? E. g. , what is lose ) almost always ( you them ' k if = { pe 1 IP ( T . so ) - Yz if ( Yp ) " p oca } I HIT l T a- { To - To ) . Then tf Hc let . of probability , formula and our continuity Now use gambler 's ruin at ppg failure - version for
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