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Frequently Asked Questions (FAQS);faqs.297 umaxc.weeg.uiowa.edu 128.255.56.80 US -6 1990/05/15 Organ: Univ. of Iowa Comment: 128.255.57.80 Files: NCSA Telnet; Sendmail umd5.umd.edu 128.8.10.5 US -5 1990/05/15 Organ: Univ. of Maryland Files: NeXT umigw.miami.edu 129.171.97.1 US -5 1990/05/15 umn-cs.cs.umn.edu 128.101.224.1 US -6 1990/05/15 Organ: Univ. of Minnesota Files: GNU; hypertext; news; Japanese umnstat.stat.umn.edu 128.101.51.1 US -5 1990/05/15 Organ: Univ. od Minnesota Files: XlispStat; S Bayes umrisca.isc.umr.edu 131.151.8.1 US -6 1990/05/16 unibi.hrz.uni-bielefeld.de 129.70.4.128 DE +1 1991/09/30 unidata.ucar.edu 128.117.140.3 US -7 1990/12/26 uniwa.uwa.oz.au 130.95.128.1 AU ? 1991/09/30 Files: RFC; Sendmail; GNU; Mac; network docs; Cnews; nn; Unix utils unix.secs.oakland.edu 141.210.180.2 US -5 1990/11/18 Organ: Oakland Univ. Files: gcc; gas; gdb; and kermit for xenix unmvax.cs.unm.edu 129.24.16.1 US -7 1990/05/15 Organ: Univ. of New Mexico Files: getmaps unx.ucc.okstate.edu 139.78.1.1 US -6 1991/02/23 Organ: Oklahoma State Univ. uop.uop.edu 138.9.200.1 US -8 1990/08/22 Files: Bnews; nn; IRC urth.acsu.buffalo.edu 128.205.7.9 US -5 1990/08/06 Files: trn usc.edu 128.125.1.45 US -8 1991/01/17 Organ: Univ. of Southern Cal Comment: 128.125.253.136 Files: AMD (automounter); DECUS TECO utadnx.cc.utexas.edu 128.83.185.100 US -6 1990/08/22 Organ: Univ. of Texas Files: VMS sources (zetaps; laser; sxlps) utnetw.utoledo.edu 131.183.1.1 US -5 1991/09/30 Organ: Univ. of Toledo Files: VMSTPC utsun.s.u-tokyo.ac.jp 133.11.11.11 JP ? 1991/05/07 Files: Japanese/Kanji software; Tron; GNU; info-mac; tex; X; network-related stuff uu.psi.com 136.161.128.3 US -5 1990/05/15 Organ: PSI Files: GNU Emacs; others; Nysernet; IETF; GOSIP uvaarpa.virginia.edu 128.143.2.7 US -5 1990/06/04 Files: u3g; whoisd; VERnet; uvapc; net docs/RFCs; misc net; etc. uvacs.cs.virginia.edu 128.143.8.29 US -5 1991/05/26 Comment: 128.143.136.10 Files: site techreports; Simple User Interface Toolkit (suit) - not the same as in X11R4; Mentat - parallel processing system uwasa.fi 128.214.12.3 FI +2 1990/12/23 Organ: Univ. of Vaasa ux.acs.umn.edu 128.101.63.2 US -6 1990/05/15 Organ: Univ. of Minnesota ux1.cso.uiuc.edu SEE: ftp.cso.uiuc.edu uxc.cso.uiuc.edu 128.174.5.50 US -6 1990/06/01 Organ: Univ. of Illinois - Urbana-Champaign Files: games; HitchHiker's Guide to the Internet; recipes; GIF; GNU; RFC; IEN; and more valeria.cs.ucla.edu 131.179.64.36 US -8 1991/11/28 Organ: Univ. of California - LA (CS Dept) Files: info-zip; zip/unzip source valhalla.ee.rochester.edu 128.151.160.11 US -5 1990/05/15 Files: RFCs; Network load balancer vax.cs.pitt.edu 130.49.2.1 US -5 1990/05/15 Files: KA9Q vax.eedsp.gatech.edu 130.207.226.2 US -5 1991/05/26 Organ: Georgia Tech vax.ftp.com 128.127.2.100 US -5 1990/05/15 Organ: FTP Software Files: FTP apps; vax1.cs.umass.edu 128.119.40.1 US -5 1990/08/22 Organ: Univ. of Massachusetts vax1.umkc.edu 134.193.1.1 ? 1990/08/22 Files: network info vax2.cs.umass.edu 128.119.40.2 US -5 1990/08/22 Organ: Univ. of Massachusetts vela.acs.oakland.edu 141.210.10.2 US -5 1991/01/17 Organ: Oakland Univ. venera.isi.edu 128.9.0.32 US -8 1990/05/15 Files: statspy (NNstat); GNU Chess venus.eng.buffalo.edu 128.205.19.6 US -5 1992/02/23 Admin: ghanem@venus.eng.buffalo.edu - Roger G. Ghanem Files: sysid apps for earthquake/structural dynamics in Fortran venus.ycc.yale.edu 130.132.1.5 US -5 1990/08/22 Organ: Yale Univ. Files: SBTeX vernam.cs.uwm.edu 129.89.9.117 US -6 1992/09/26 Organ: Univ. of Wisconsin - Milwaukee Files: 386BSD source & binaries; Suni386 binaries and mailing list; Mach386 binaries vesta.sunquest.com 192.12.52.42 ? 1990/08/22 Comment: 149.138.1.42 Files: VMS cryptodisk and virtual disk driver vlsi.cs.umn.edu 128.101.230.15 US -6 1990/08/22 Organ: Univ. of Minnesota Files: PLP (Public line printer) vm.tcs.tulane.edu 129.81.128.1 ? 1990/08/22 Files: RFC; terminfo vm.utdallas.edu 129.110.102.2 US -6 1990/08/22 Files: lipphe IBM TCP/IP bit and drivers vm1.nodak.edu 134.129.111.1 US ? 1990/08/22 Files: Minix fixes; EV1188 fixes; QNS; medical newsletter vmd.cso.uiuc.edu 128.174.5.98 1990/12/24 ---> REMOVED 1992/09/10 vms.ecs.rpi.edu 128.113.5.15 US -5 1990/08/22 Organ: Rensselaer Polytechnic Institute Files: Message Exchange vms.huji.ac.il 128.139.4.3 IL +2 1991/09/30 vms2.ecs.rpi.edu 128.113.5.14 US -5 1990/08/22 Organ: Rensselaer Polytechnic Institute Files: NEWSRDR vmsa.oac.uci.edu 128.200.9.5 US -8 1991/02/21 Organ: Univ. of California - Irvine Files: VMS make; TAR vmsa.technion.ac.il 132.68.1.40 IL +2 1991/09/30 vmsd.oac.uci.edu 128.200.15.2 US -8 1990/08/22 Organ: Univ. of California - Irvine Files: VMS stuff vmtecmex.cem.itesm.mx 132.254.1.4 MX ? 1991/02/08 Files: Amiga; MSDOS; GIF walhalla.informatik.uni-dortmund.de 129.217.64.63 DE +1 1991/11/28 watcgl.waterloo.edu 129.97.140.64 CA -5 1990/10/09 watmath.waterloo.edu 129.97.140.144 CA -5 1990/05/15 Comment: 129.97.216.42 Files: lots of stuff watmsg.waterloo.edu 129.97.141.9 CA -5 1990/05/15 Files: GNU; pd BSD; uSystem docs; virus; cryptography watnxt2.ucr.edu 192.31.146.189 US -8 1990/10/09 Organ: Univ. of California - Riverside watserv1.uwaterloo.ca 129.97.129.140 CA -5/-4 1992/07/29 Organ: Univ. of Waterloo Files: APL stuff in /languages/apl watserv1.waterloo.edu 129.97.129.140 CA -5 1990/08/22 watsun.cc.columbia.edu 128.59.39.2 US -5 1990/05/15 Organ: Columbia Univ. Files: kermit wayback.cs.cornell.edu 128.84.254.7 US -5 1991/04/24 Organ: Cornell Univ. weedeater.math.yale.edu 130.132.23.17 US -5 1990/05/15 Organ: Yale Univ. Files: rayshade; misc. raytracing goodies white.cerritos.edu 130.150.200.22 ? 1991/09/30 Files: rec.motorcycles.pictures wilbur.stanford.edu 36.14.0.30 US -8 1990/12/26 Organ: Stanford Univ. wilma.cs.brown.edu 128.148.31.66 US ? 1991/09/19 Files: Brown CS Field and Thread packages; comp.robotics; XMX wlv.imsd.contel.com 192.26.147.1 US -5 1991/05/26 Organ: Contel Corp Comment: 26.5.0.103 wolfen.cc.uow.edu.au 130.130.68.4 AU ? 1991/01/17 wowbagger.pc-labor.uni-bremen.de 134.102.228.9 DE +1 1992/06/29 Organ: Univ. of Bremen Files: Amiga; Atari-ST; X11; TeX; neiral-nets; GNU (mirror of prep.ai.mit.edu) wpi.wpi.edu 130.215.24.1 US -5 1990/05/15 Files: dspl; anime; fusion; Mac; GNU; ASH; DES; misc Unix; TeX_DS3100; TeX_Umax; misc X wsmr-simtel20.army.mil 192.88.110.20 US -7 1992/09/16 Organ: U.S. Army - White Sands Missle Range Files: MSDOS; Unix; CPM; Mac (tenex) wuarchive.wustl.edu 128.252.135.4 US -6 1992/09/16 Organ: Washington Univ. Comment: Open 24 hours Files: GNU; X.11R3; GIF; IEN; RFCs; TeX; UUPC; info-mac; 4.3BSD-Tahoe; comp.binaries: amiga, apple2, atari.st, ibm.pc; comp.sources: amiga, games, misc, sun, unix, x; msdos; Archives - SimTel20, cica (as mirrors/win3); Elm xanth.cs.odu.edu 128.82.8.1 US -5 1990/08/01 Comment: 128.82.4.1, 128.82.4.65 Files: comp.sources: games, misc, x, unix xanthorrhoea.maths.uwa.oz.au 130.95.16.13 AU ? 1991/09/26 xcf.berkeley.edu SEE: scam.berkeley.edu xview.ucdavis.edu 128.120.1.150 US -8 1990/07/19 Organ: Univ. of California - Davis Files: xview yalevm.ycc.yale.edu 130.132.1.4 US -5 1990/08/22 Organ: Yale Univ. yallara.cs.rmit.oz.au 131.170.24.42 AU ? 1991/01/17 Comment: 192.55.190.42 ymir.claremont.edu 134.173.4.23 US -8 1990/08/22 Organ: Claremont College Comment: 134.173.20.23, 134.173.21.23 Files: lots of VMS; TeX-for-VMS; mainz fonts z.eecs.umich.edu SEE: ftp.eecs.umich.edu zaphod.lanl.gov 128.165.44.202 US -7 1991/02/23 zaphod.ncsa.uiuc.edu 141.142.20.50 US -6 1990/08/13 Organ: Univ. of Illinois - Urbana-Champaign Files: Mac X aaps; NCSA Telnet; Heirarchical Data File system; misc graphics & scientific formats zariski.harvard.edu 128.103.28.10 US -5 1990/05/15 Organ: Harvard Univ. Files: macaulay zerkalo.harvard.edu 128.103.42.201 US -5 1990/05/15 Organ: Harvard Univ. Files: traceroute zeus.cs.umu.se 130.239.32.12 SE +1 1991/11/28 Comment: 130.239.80.2 zeus.mgmt.purdue.edu 128.210.1.3 US -5 1990/05/15 Organ: Purdue Univ. zeus.unomaha.edu 137.48.1.1 US -6 1991/02/21 Organ: Univ. of Omaha Files: cedit - column editor Path: bloom-picayune.mit.edu!snorkelwacker.mit.edu!hri.com!spool.mu.edu!agate!usenet.ins.cwru.edu!magnus.acs.ohio-state.edu!csn!hellgate.utah.edu!hellgate!jacobs From: jacobs@cs.utah.edu (Steven R Jacobs) Newsgroups: rec.gambling,news.answers Subject: rec.gambling Changes to Frequently Asked Questions Date: 5 Oct 92 02:09:43 GMT Reply-To: jacobs@cs.utah.edu Followup-To: rec.gambling Organization: University of Utah CS Dept Lines: 110 Approved: news-answers-request@MIT.Edu Expires: Fri, 13 Nov 1992 08:00:00 GMT Archive-name: gambling-faq/diff Comment: A blank line that was in the original posting above was deleted by the maintainer of the pit-manager.mit.edu FAQ archive, to bring the posting into a consistent format. Note: This "diff" file has been edited and is not intended for use by any automatic update software. --- > G10 What are some cheap places to stay in Vegas? --- > B14 Why are single deck games better than multi-deck games? --- > B23 What is the correct strategy for "multi action" blackjack? 513,514c516,517 --- > Q:G10 What are some cheap places to stay in Vegas? > A:G10 (Ron Birnbaum) > > In response to those looking for nice but inexpensive accomodations in > Vegas, I can recommend two places: > > Downtown: The Ogden House - Fremont Street > I don't have their number, but some friends stayed there in > June for $18 per night, and found it clean and comfortable. > Their phone number is (702) 385-5200. > > Strip: King Alburt Motel > It's behind the Flamingo Hilton/Imperial Palace/Barbary Coast area. > Every unit is furnished with a kitchen, tv, and private bath, with > simple but clean furnishings. Their location is ideal, and they > quoted me day rates of $25/wk $40/wkend per night for a single. > And these prices were for March,('92) - usually a busy and high > priced season. And you can do even better than these rates for a > weekly rental. They are happy to negotiate! I don't believe they > have a toll-free number, but I'm not sure. Their regular number is > (702) 732- 1555 > > Call the Las Vegas Tourism Beareau for details of other low-cost > area lodging. They have an toll-free number - 1-800-522-9555. Woops - I > just called it and they only handle casinos. > > But these 800 numbers DO handle the smaller places as well. I called the > LV Chamber of Commerece for these numbers, and they had even more than > these: > > 800 - 548-2008 > 800 - 345-7177 > > Through the 7177 number, I arranged a room in the beautiful new tower at > the Stardust for much lower than what the hotel quoted me directly, also > at a lower rate than the other 800 numbers could offer (32/wk 40/wkend). > They are a room 'wholesaler' and seem to be the best place to make > reservations through. The new tower at the Stardust is supposed to > be ****1/2, according to Julie - the agent I spoke with. She also asked > for a high floor which she said have spectatular views of the entire strip. > > They know all the deals in town! Good luck, hotel bargain hunters!! > 1138,1139c1185,1186 --- > Q:B14 Why are single deck games better than multi-deck games? > A:B14 (Michael Hall) > > There are some surface differences, such as single and double deck usually > being hand-held, while four or more decks are dealt from a shoe, but there > are fundamental mathematical differences too. > > Single deck blackjack is usually better than multiple deck blackjack > for card counters, basic strategists, and the clueless. Additional decks > make busts less likely, since one can draw to hands like 2,2,2,2,2,2,2,2,2 > (for 18) which are improbable/impossible in single deck. Busting less > often helps the dealer's hand more than yours, since the dealer is forced > by the rigid rules to hit more often than you. Blackjacks are also less > frequent, which is bad since you get paid 3 to 2 for those. All in all, > multiple decks will cost a basic strategist nearly 0.5% in advantage, which > is more than all but the very best package of favorable extra rules will > give you. This was an intuitive explanation; a complete mathematically > sound (albeit huge) proof can be generated by a combinatorial analysis > program. > > Card counters face the additional problem that the count is less volatile > with multiple decks and hence offers less frequent opportunities for large > favorable bets. Consider the difference between an urn with 1 black and > 1 white marble versus an urn with 100 black and 100 white marbles. Draw > half the marbles: what is the probability that all the remaining marbles > are white? In the 1 and 1 case, there is a 1 in 2 chance. In the 100 and > 100 case, there is only a 1 in 100,891,344,545,564,193,334,812,497,256 > chance! > > 1392,1393c1469,1470 > Q:B23 What is the correct strategy for "multi action" blackjack? > A:B23 (Steve Jacobs) > > Multi Action blackjack allows the player to place up to three bets > simultaneously on the same blackjack hand. The player is dealt a > single hand, and the three bets are played out against the same dealer > upcard, but with different "drawn" cards for each bet. Many players > feel nervous about hitting stiff hands against a high dealer's upcard > (7 or higher), since they will lose all three bets if they bust. > However, basic strategy is COMPLETELY UNCHANGED for this game, and the > correct strategy is no different than if the player had only a single > bet at risk. > -- Steve Jacobs ({bellcore,hplabs,uunet}!utah-cs!jacobs, jacobs@cs.utah.edu) "Be afraid. Be very afraid. I have K-10 offsuit...." Xref: bloom-picayune.mit.edu rec.gambling:12274 news.answers:3399 Newsgroups: rec.gambling,news.answers Path: bloom-picayune.mit.edu!snorkelwacker.mit.edu!news.media.mit.edu!micro-heart-of-gold.mit.edu!wupost!zaphod.mps.ohio-state.edu!caen!hellgate.utah.edu!cs.utah.edu!jacobs From: jacobs@cs.utah.edu (Steve Jacobs) Subject: rec.gambling Frequently Asked Questions Date: 8 Oct 92 10:45:24 MDT Message-ID: <1992Oct8.104525.13806@hellgate.utah.edu> Followup-To: rec.gambling Lines: 2470 Reply-To: jacobs@cs.utah.edu Approved: news-answers-request@MIT.Edu Expires: Fri, 13 Nov 1992 08:00:00 GMT Archive-name: gambling-faq/part1 ================================== This is the rec.gambling Frequently Asked Questions (FAQ) list. I wish to thank Frank Irwin for helping to compile much of the information that appears in this list. I also wish to thank those who contributed information, as well as those who spent countless hours running blackjack simulations in order to help answer many of these questions. Changes or additions to this FAQ list should be submitted to: jacobs@cs.utah.edu ================================== Table of Contents Section S: Spare us! S1 Martingale betting systems -- just double your bet until you win S2 The Inevitable Monty Hall S3 How much would you pay to play this game? Section G: General Gambling Topics G1 What's in it for me? (Why read rec.gambling) G2 Where can I get books about gambling? G3 Is there a gambling archive? G4 How do you get comps? G5 What comps are available? G6 How do I get a casino credit line? G7 How are "markers" used? G8 What are the phone numbers for hotels/casinos in Las Vegas? G9 What are some good places to play/stay in Reno? G10 What are some cheap places to stay in Vegas? G11 Where can I get casino quality chips? G12 What are matchplay chips G14 Is this Vegas World offer worth it? G15 When did rec.gambling get started? Section B: Blackjack B1 What do these funny acronyms mean ... B2 What special terminology is used by blackjack players? B3 What special terminology is used by card counters? B4 What are "pit critters"? B5 Why is there so much talk about blackjack in rec.gambling? B6 Is casino blackjack a "beatable" game. B7 How much of an advantage can card counting give? B8 Is card counting illegal? B9 Can the casino ban card counters? B10 What is the correct basic strategy for single deck Blackjack? B11 What is the correct basic strategy for Atlantic City blackjack? B12 What is the house edge when playing basic strategy? B14 Why are single deck games better than multi-deck games? B15 Do 'bad' players at third base have any effect on expected gain? B16 Where is the best place to sit at a blackjack table. B17 How is card counting done? B18 What counting system is "best"? B19 What counting system is easiest to use? B20 What BJ counting system is most effective? B21 Does penetration have any effect on basic strategy expectation? B22 What is the correct strategy for late surrender? B23 What is the correct strategy for "multi action" blackjack? B24 What is "Over/Under" Blackjack? B26 What is the counting strategy for Over/Under blackjack? B26 What are some good/bad books on Blackjack? B27 What are some other sources of blackjack/gambling information? Section C: Craps C1 What special terminology is used at the Craps table? C2 How is Craps played? C3 What are "Odds?" C4 What are "Come" and "Don't Come" bets? C5 What are all those other bets? Section V: Video Poker V1 Is it possible to gain an advantage at Video Poker? V2 What is the "basic strategy" for Video Poker? Section P: Poker P1 How is Texas Hold'em played? Section M: Miscellaneous M1 How is Baccarat played? M2 How is Red Dog played? M3 Can the lottery be beat when the jackpot gets high enough? M4 How is Pai Gow Poker played? M5 Is there a horse racing newsgroup? ================================== Section S: Spare us! These questions come up occasionally, causing a lot of heated discussions and wasted bandwidth. The rec.gambling regulars are pretty sick of seeing these questions, and they would appreciate it if you just didn't ask them. If you do ask, we'll probably just say "see the FAQ list". Q:S1 Martingale betting systems -- just double your bet until you win A:S1 (Frank Irwin, Steve Jacobs) From: "The Eudaemonic Pie" by Thomas A. Bass The word comes from the French expression "porter les chausses a la martingale," which means "to wear one's pants like the natives of Martigue," a village in Provence where trousers are fastened at the rear. The expression implies that this style of dress and method of betting are equally ridiculous. The betting scheme merely states that you would want to double your bet after each loss. Beginning with one unit, you would bet two units if you lost the first. Then four, then eight, until you win a bet. You would then revert to a one unit bet. The theory is that with each win you will win all that you lost since the last win, plus one unit. The reality is that you will quickly come to a betting ceiling, governed by either your bankroll or the house limit, above which you may not increase your bet. After 9 straight losses (it's happened to me) you would be betting 512 units. In practice, a lot of people get sucked into betting this way because it gives the illusion of really working. This is because most of the time, you will end a string of bets with a win. However, on those rare occasions when you do lose, you will lose a lot of money. So, the end result is that you win a small amount almost always, but when you lose you will lose more than all of your little wins combined. The important point to realize is that most games simply cannot be beat in the long run. In games such as craps, roulette, and non-progressive slot machines, it is mathematically impossible to gain an advantage over the house. Q:S2 The Inevitable Monty Hall A:S2 (Steve Jacobs) You are a contestant on "Let's Make A Deal", and Monty Hall offers you your choice of three doors. One door has a prize, and the other two doors are empty. Monty knows in advance where the prize is, and no matter which door you pick, Monty will open one of the other doors to show you that it is empty. Monty then offers to allow you to trade your door for the other unopened door. What should you do? My best advice is to grab Monty around the neck and strangle him :-) The short answer is that you should switch doors, because it increases your chance of winning from 1/3 to 2/3. The odds are not 50/50, because the probability that the prize is behind the door you originally picked is 1/3, and this probability never changes until the prize is revealed. This can be seen by considering what happens if you decide to *never* switch doors -- in this case you will win one time in three, since you will win only if your original pick is correct. So, if you *always* switch you will win 2/3 of the time, since you win whenever your original pick was wrong. Since Monty can (and will) always show an empty door, it is as if he is saying "you can keep your door, or you can trade is for BOTH of the other doors, and to confuse you I will show you that one of the other doors is empty, even though you already knew that". Some of you won't believe that this is correct. You will say "after Monty opens a door, there are only two choices so you have an equal chance of winning whether you switch or not". This is wrong. The fact that there are only two choices does NOT imply that the two choices have equal probability. Still not convinced? Suppose there are 100 doors, and only one prize. You pick a door, and Monty shows you 98 empty doors (he can always do this, since he knows where the prize is), and offers to let you switch. The chance that your original pick was right is 1/100. If you never switch, you will almost always lose. Therefore, if you always switch you will almost always win. Not convinced? Try it yourself, but try playing the role of Monty in order to help see how it works. Don't ask rec.gambling, because we'll know you didn't really try it yourself. The confusion caused by this question shows that probability problems often go against human intuition. This question appeared (several times) in the "Ask Marilyn" column of the "Parade" insert that appears in many Sunday newspapers in the United States. Many people with Ph.D.'s in mathematics claimed in wild disbelief that Marilyn's answer [2/3] was wrong, and that it was a sad commentary on the American education system. Indeed it is, because the mathematicians were wrong. Q:S3 How much would you pay to play this game? A:S3 (John P. Nelson) >I flip a coin until it comes up tails. If it comes up tails the first >time, you get $2; if the second, $4; if the third time, $8. That is, >if I flip the coin N times, you get 2^N bucks. [How much would you >pay to play this game?] The problem is, of course, that the "expected value" of this game is an infinite series that does not converge: The "expected" win appears to be infinite. However, in any REAL implementation of this game, there would have to be an upper limit to your liability: It would be unreasonable to expect you to really pay off a win of trillions of dollars. If you put an upper limit on the number of flips, then the expected value of the game converges, and it is quite easy to calculate the "break even" wager. If we limit the game to N flips, and you pay me $2^M if tails comes up on the M'th flip and if N flips occur with no tails, you pay me 2^(N+1), then the expected value of each game is simply N+2. (A variation on the rules: If you pay me 0 if N flips occur with no tails, then the expected value of the game is $N). This is fairly easy to calculate: On each independent trial, the chance of a tails is 1/2. Clearly, the chance of a tails on the second trial is 1/4 (1/2 chance of heads on the first trial, times 1/2 chance of tails on the second trial). The odds of a tail occuring on the Nth flip is 1/(2^N). The payoff at that point is 2^N. So we have the series: 1/2 * payoff-1flip + 1/4 * payoff-2flip + 1/8 * payoff-3flip or: $2 $4 $8 $(2^N) -- + -- + -- ... ------ 2 4 8 $(2^N). Given a maximum of N flips, the expected win is 1+1+1..., N times. Given a $0 payoff on N heads in a row, that is the total expected win: N. If you pay off 2^(N+1) after N tails, the final term is $(2^(N+1)) ---------- = 2 $(2^N) Which explains where the +2 in (N+2) comes from. ==================================