chapkrt
play

Chapkrt win ? can I : we distribution ps.iq So with iid Last time - PowerPoint PPT Presentation

Chapkrt win ? can I : we distribution ps.iq So with iid Last time : { G) new let be . at ?g Define " hitting tines " - 1 let 2 Ci Xu . - O } , = inf { n > O : Xu - c } and To - int { n 30 - : Xn Te - - .


  1. Chapkrt win ? can I : we

  2. distribution ps.iq So with iid Last time : { G) new let be . at ?g Define " hitting tines " - 1 let 2 Ci Xu . - O } , = inf { n > O : Xu - c } and To - int { n 30 - : Xn Te - - . ±um:i:÷ini÷¥÷:::: - Ta ) =/ peek PIT . - Yz p - c

  3. - 0.49 Ex - p :f÷:f""÷÷n a - i 1,8% 100000 99900 900001000002 × 10-17+17 The $1000 but in to size increase your if keawwg odds of winning you : increase your significantly you last case , you're in the , time at $10000 a If yer bet chance ! 88% scenario → 9 , - lo a- c -

  4. tMf coin flip game the In , we charge our can to strategy gambling " ? " do better

  5. Dein ( Gambling Policy ) last time , let nothin from the keeping - at .EE?Y::.Imae Xu - " win " of ith of " losing " bet or - fi . ! 9,4 , - Ici - i ) where and Wi 30 Wi - - .

  6. Play ) Exe ( Bold dollars let reach want to c . Suppose we - Xin ) min ( Xi - i , c Wi = . without going over c most you idea : bet the . can maximizes strategy this Deep theorem : IPL To < To ) .

  7. win ) ( Double til you EI Wi . { - Ci . , - o " ' it c. - 2 = - - - and W , =L , let else O - Xs = Xz=Xs : Xy - - if - = att = - 4--1 X then , , = att - a -1+2 and -_ a - I Xz X , - , then else cut - Xy - Ys - X , = - - - - - - - a -3 - a - l 4=1 , then X Xa= a - I -2 - else - , , , = Xie - att = Xs - 31-4 ± Xy : - - - X , - a = - Xut X.at ? : Cn -4 ) - influx N , Then Xn att if check N' - you can

  8. get this scenario we Note in , - att ) - I IP ( " Y X n - - " double til this that Question does mean c this game ? . winning strategy in gives " win a you - IE ( att ) IE ( " nm Xn ) at I - - aren't deep pockets , because Answer your no enough : .

  9. azitzt-e.tk with integer the largest be let KEN " bdovble lil you times ) win " K play lie , can we - MEK inffnza.cn if have - X µ = att we if iuxffn > : Cal ) > k - 2K - I - 2- else Xk - - - - a . - qkfa - qk ) ( att ) - 2K ) th ⇐ ( Xk ) - I -2 - we - - - - see - 2(2q)k = att pet if than less This a is .

  10. doesn't actually til you win double We've seen system ? will " beat " any the casino . ÷2 Wilhoit ) = at Recall Xu , = at II - i ) wi @ Ci - t ) t Wn @ Cn , - 1) Xu , t Wn ( kn = - i ) - tn ( Ci , . - , Cn where knew Wn But - we independent . . - - , Cn - i , Cn C are .

  11. independent , 2cm - I so and are So Wn ⇐ ( Xu - it win @ Cn - t ) ) ⇐ ( Xn ) = - t ) ) ⇐ ( Wn @ en . . ) IE ( Xn t = ⇐ ( Wn ) ⇐ ( kn - t ) IE ( Xu - i ) t = - q ) IELWN )lp " IE ( Xn . . ) t - 30 - IEC # So : if a - IECX , ) then - I - - - = - p - , a > IECX , )3IECXz ) 's if petz , then - - - ( X. Is # ( Xz ) E if z , thin - as - - ' p >

  12. if bum IE( Xn ) sa pet this gives Nate . win double til you saw , with , we contrast In " nm Xn ) - att ⇐ ( - . - tellin Xa ) ? by # Xn ) when - is MCT of EXIT , then know I ;mXn We nm Xa ) ⇐ ( " linm # ( Xn ) says - . -

  13. ( Bounded convergence Theorem ) Thou almost surely . If Ippon exists lynx . that , then IX. Is k have all for we n KEIR so " nm Xu ) E- ( nm Elk ) " = . say , for ;4 The triage inequality - X PI Dude in Xu - . - Xml ) E IE ( IX - Xn ) / I ⇐ ( X - IE ( Xn ) I . I ⇐ ( x ) = so for all NEIN have all : for e > 0 we want We - E- ( Xn ) l K E IIECX ) we get . n z N - Xal ) - a -50 s O El IX as do this shewing by . we

  14. - Xnlwll > e } - { wer I Xlw ) Define let E > 0 An : - . . have all we .r have for we We - Xnlw ) Is et 2K Han ( w ) I Xlw ) expectations Take IEC et 2K Iancu ) ) - Xnlw ) l ) E ⇐ ( lxlw ) et 2K IPC An ) = sides both take kmsup Naw on " TYP MAN ) set 2K Pl ' 'm :P An ) - Xml ) Et 2K ⇐ ( IX " map E

  15. - X. Cull > e ) - " must { wer : lxlw ) An limsup But - line Xnlw ) t Xlw ) with of , is the set w DNE ) { wer : hnmxnlw ) just which is . is measure zero . set hypothesis this by But Hence " TYP MANI ' 'm :P An ) ' et 2K Pl - Xml ) Et 2K ⇐ ( IX " mhm E - E E . " I'm IEHHXND fer " msn.PE/lX-Xal ) < E all s , s . S . = Off

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend


More recommend