posted
OK, so there's a three way duel. Each dueler gets to shoot once at any other dueler. If he misses, the next dueler can shoot at any other dueler. The duel continues until only one man is left standing. Dueler #1 hits one out of every three shots he takes. Dueler #2 hits two out of every three shots. Dueler #3 hits three out of every three. It is decided that Dueler #1 will shoot first, Dueler #2 second and Dueler #3 third.
If Dueler #1 has any chance of survival, who should he shoot at first?
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posted
For any chance of survival, #1 should shoot at #3 and miss, so that #3 will shoot and hit # 2 and pray that #2 misses so he can have counselor Troy all to himself
Posts: 26 | Registered: Jun 2003
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posted
I remain to be convinced by the 'shoot at #3 and miss' argument - if #1 misses, it doesn't matter who he was shooting at.
Assuming that we are dealing with simple probabilities here (#1 has a 1/3 chance to hit with every shot, independently of what has happened before), rather than some situation where he hits on every third shot (2 guaranteed misses, one guaranteed hit).
It's pretty easy to draw up a decision tree for this game. You only need one that goes up until the first hit is made, which at worst is going to be when #3 fires his first shot. After the first hit we have 6 possible two way duels, the probabilities of which are known exactly. Using the notation A vs B as meaning A and B are in a two way duel with A shooting first, the probability of A (the first shooter) winning the 2 way duel is: #1 vs #2 : 3/7 #1 vs #3 : 1/3 #2 vs #1 : 6/7 #2 vs #3 : 2/3 #3 vs #1 : 1 #3 vs #2 : 1
Whenever we get to #3 firing and having a choice of targets, he could shoot #1 putting him in a #2 vs #3 duel (which he will win 1/3 of the time), or he could shoot #2, putting him in a #1 vs #3 duel (which he will win 2/3 of the time). So he should always shoot #2 to maximise his chances.
When #2 gets a choice of targets, he knows, by the above logic, that #3 will shoot him unless he kills #3 now (if he shoots #1, he ends up in a #3 vs #2 duel which he is certain to lose), so he targets #3 every time, no matter what #1 did.
So if #1 misses, the outcome doesn't matter on who he shot at - #2 will shoot at #3, and #3 will kill #2 if he survives. #1 wins a total of 50/189 duels when he misses.
If #1 shoots at #2 and hits, he is in a #3 vs #1 duel which he is certain to lose. If he shoots #3 and hits he is in a #2 vs #1 duel, which he wins 1/7 of the time. That's an extra 9/189 wins for #1 if he shoots #3.
Net result: if #1 shoots at #2 he has a 50/189 chance of winning the duel. If he shoots at #3 he has a 59/189 chance of winning. He should shoot #3.
posted
And then I spot the trick to this: #1's chances are best when he misses, and worse when he hits someone (when he hits, he is always the person who fires second in the duel, giving him a 1/7 chance or a 0 chance, depending on who survives out of #2 and #3).
So #1 fires at a nearby tree instead (and hits it 1/3 of the time ) #2 fires at #3, hitting 2/3 of the time. So 2/3 of the time we end up in a #1 vs #2 duel which #1 wins 3/7 of the time. 1/3 of the time we end up in a #1 vs #3 duel, which #1 wins 1/3 of the time. This makes #1's total chances of winning 25/63 (= 75/189), which is better than his odds for aiming at either opponent.
Posts: 9 | Registered: Aug 2003
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posted
Here's another mathematical-type puzzle, although one that some of you may already be familiar with.
In a game show, there are three large boxes. One of them is known to contain a car, while the other two each contain a goat. The rules of the game are these: You have to pick a box. The game show host will then open the door to one of the boxes that you didn't pick, to show you that that box contains a goat (which the host can always do). You then have the choice of sticking with your originally chosen box, or switching to the other unopened box. You win the car if you pick the box containing the car.
Does it make a difference which of the two unopened boxes you pick, and if it does, which box is it better to pick (i.e. is it better to stick with your original choice, or swap to the other box)?
Unlike the first puzzle, there isn't really a trick to this one. Assume that there is no way to determine what is in the boxes aside from opening them, and that the host always has to open a goat box (and doesn't just to it when you have picked the car box, for instance).
Actually, the fun with this one isn't in trying to get the answer, it is in trying to explain to the people who refuse to believe it why it is right...
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posted
I remember this problem. I remember that the answer was that you should switch. I even remember that I agreed that you should switch. But I can't for the life of me figure out, now, why it should matter which box you pick.
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posted
Oooh. Sorry I was late for that classic original problem.
Unfortunatly, probability says that if #1 has 1/3 chance, every shot that he makes still has the same odds. So it could actually be that #1 could, unlikely though it may be, shoot them all dead right away, but he would have to get a shot in before the guy that won't miss!
posted
The little alien woman from Mars has even chances as long as she has no information about which box you originally picked. If she knows that, then obviously she has the same chances as you.
Posts: 9 | Registered: Aug 2003
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posted
Basically, the first round of that game show doesn't matter. Regardless of what happens during that round, it comes down to a fifty-fifty choice in the second round, with the irrelevant knowledge that there is an exposed goat standing nearby. Someone's going to have to work very hard to convince me otherwise.
Wait a minute ... [thinks hard] ... at the outset, there is a 2/3 chance that you picked a goat, right? ... hmm ... So actually, yeah, since your initial pick is based on a three-way choice, it IS less likely to be correct than a choice made with 50-50 odds. Whoah. Weird. Okay, I buy it.
Posts: 2048 | Registered: Jul 2000
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posted
i've never heard this problem before but, isnt the dilemma Choosing between Logic and Spirituality? As in, Do i sied with probability and switch to the middle box or stay with my gut instinct that told me right away to choose the box on the far left. I am very curious as to what the explanation is to both of these problems because my brain hurts right now from. My only conclusion as far as #1 is concerned is that he should take whatever routes grants #3 the least opportunity to fire since he's more or less guaranteed to hit. yeah ..that's it. Posts: 26 | Registered: Jun 2003
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posted
i apologize for the typos. I'm at work and share an office with my boss and have to sneak these replies in.
Posts: 26 | Registered: Jun 2003
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Flipping a coin produces a 50% chance that it will land tails, 50% that it will land heads.
If you flip a coin 99 times, and it comes up heads every one of those 99 times, then the odds that it will come up heads on the next toss are still 50/50.
Posts: 2506 | Registered: Jul 2003
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posted
It was unlikely in the first place. And still is, but it is not cumulative.
Edit: Boy was my statement messed up or what? It isn't unlikely because the odds are always the same whenever you toss the coin. It's just not easy to wrap your brain around (not easy for me, at least). Everyday experience seems to contradict it, but if it happens, and probably must happen eventually, then it still obeys those same laws of probability.
quote:If you flip a coin 99 times, and it comes up heads every one of those 99 times, then the odds that it will come up heads on the next toss are still 50/50.
If you flip a coin 99 times, and it comes up heads every one of those 99 times, then the odds are that it's a two-headed coin, and there's a 100% chance it will come up heads on the next toss.
Here is the statitic question that I don't have the answer too:
If you go to Vegas, or anywhere they gamble, you will have a chance to play a Lotter based game who's name is slipping my mind.
You are trying to pick 10 numbers from a field of 70 or some such.
The house has even gone so far as to display for you how often certain numbers were called in the last 50 times.
1 = 1 time 2 = 5 times 3 = 30 times 4 = 0 times...
The question is, is it better to:
A) Pick numbers that have won often, for they are favored. B) Pick numbers that have won few times, for they are due to come up. C) Ignore the numbers. In fact, take your money to the Black Jack tables instead.
posted
The game is Keno. One chooses from one to ten numbers, then those are matched against a set of twenty numbers from one to eighty, and payouts are determined through a combination of how many you matched out of how many you chose. In a perfectly random system, head to blackjack, 'cause the odds are better if you know what you're doing (and the previous draws make no difference in Keno). However, most Keno games are played by a computer (real people couldn't draw 20 numbers every three-and-a-half minutes and stay organized), which is only pseudo-random. In the short-term (last 50 draws falls into that category), you want to pick the numbers that come up less frequently, because the nature of what the computer considers randomness means every number has approximately the same chance, and it's really cycling through a huge set of numbers. In the long-term (probably at least in the billions of draws by the computer), choose the numbers that come up most frequently, because they appear more often in the pseudo-random cycle.
posted
Dan, as someone who gambles often, and has dabbled in roulette here and there, i gotta say that that's an extremely good question.
I'm always torn between betting against the streak(because obviously it's due to come up ) or to just bet with the streak.
I solved this problem by not playing roulette anymore. Just blackjack and poker for me thank you. So i guess(if your question carries over to roulette as well) option number 3.
quote:Assuming that we are dealing with simple probabilities here (#1 has a 1/3 chance to hit with every shot, independently of what has happened before), rather than some situation where he hits on every third shot (2 guaranteed misses, one guaranteed hit).
Your assumption is wrong...
quote:So #1 fires at a nearby tree instead (and hits it 1/3 of the time ) #2 fires at #3, hitting 2/3 of the time. So 2/3 of the time we end up in a #1 vs #2 duel which #1 wins 3/7 of the time. 1/3 of the time we end up in a #1 vs #3 duel, which #1 wins 1/3 of the time. This makes #1's total chances of winning 25/63 (= 75/189), which is better than his odds for aiming at either opponent.
But this is correct.
Dueler #1 is guaranteed 1 hit and two misses for every three shots. It's a closed system. That's why I didn't say that he hit one-third of his shots. So, if he misses intentionally on his first shot his next shot will have a 50% chance. Dueler #2 will still shoot at #3. If he hits #3, #1 will shoot at him. If #2 misses #3, #3 will shoot #2 and #1 will shoot at him.
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quote: If you flip a coin 99 times, and it comes up heads every one of those 99 times, then the odds that it will come up heads on the next toss are still 50/50
Eslaine's point here was that if each event (coin flip or Keno drawing in Dan's example) is independant, then past performance is irrelevant, in the Keno game or the coin flips. Papa Moose makes a good point: a coin that flips heads 99 times is almost certainly either double-headed or at least weighted to generate heads. The odds of a good coin flipping like that is 2^-99=6x10^-29, a vanishingly small number. You are more likely to win most lotteries many times before you'd get 99 heads. It is possible, just incredibly unlikely.
Most people would say that a tail is more likely after 9 (lets make it believable) heads. This is known to mathematicians as "the Gamblers fallacy." If the events are independent, the odds for any coin flip with a good coin will always be 50-50 no matter what flips occured in the past. It does seem counter-intuitive.
Blackjack is different. Each draw effects later draws. This is why card-counting (different techniques, but basically keeping track of 10-value cards versus other cards) in Blackjack improves your odds. In the old days, before card-counting became more common, Vegas and Atlantic City casino's used to use one or two deck shoes to deal Blackjack. With a one or two deck shoe, card-counting really boosts your odds. So most modern casinos use 5 or more deck shoes to deal Blackjack, and shuffle well before the shoe is empty. Card-counting against this system only gives a slight 1 or 2% gain, hardly worth the trouble. I have heard that some Indian casinos use a 2 or 3 deck shoe for Blackjack, cannot confirm. Also, card-counting is legal, as long as you only use your brain, no artificial aids. But casinos that suspect you of it will toss you and blacklist you. After all, you're there to lose (from their viewpoint.)
[edit to add: Pop makes another good point about Keno: numbers chosen by computer are usually pseudo-random, not truly random. So the drawings in Keno may or may not be independent. Randomness is surprisingly difficult to give a rigourous mathematical definition for.]
A) Pick numbers that have won often, for they are favored. B) Pick numbers that have won few times, for they are due to come up. C) Ignore the numbers. In fact, take your money to the Black Jack tables instead.
The gamblers fallacy applies here too. From a strictly statistical sense, it makes no difference what numbers have been picked before for different draws are independent of each other. In fact, Vegas knows this and that is why they give you the numbers in the first place. Your best bet in this game would be to take your money and go play poker or blackjack. Those games are two of the most winnable games if you know what you are doing.
Edit: I wanted to add the probability of picking all 10 numbers out of a lot of 70. It is about 1 in 250 billion. Good Luck.
quote:Card-counting against this system only gives a slight 1 or 2% gain, hardly worth the trouble.
ahhh...Morbo, I beg to differ. and you were doing so well! I agreed with everything else in your post.
In the long run, and that is the important part of this, a 1 or 2% gain is without a doubt worth the trouble. Your odds in blackjack are so close to 50% normally, but sadly just under, that while playing by the book will increase your odds of doing better or "losing less money, in the long run you will eventually lose. That slight increase, which you say is not worth your trouble, puts you up over 50%, so that in the long run the odds are in your favor to win.
quote:Your best bet in this game would be to take your money and go play poker or blackjack. Those games are two of the most winnable games if you know what you are doing.
Jerry, actually Bacarat has the best odds of any casino "table" game if you know what you're doing. Apparently video poker also has very good odds for a player that knows what they're doing, but i've never really looked into it. But from what i know there is a system that if followed pays out well.
also, just to note, vegas still has some casinos that have one and two deck blackjack. but it's played a little differently, players keep their cards face down and only flip them after the hand ends(goes around the table), and i think you lose on a push, which sucks.
quote:go play poker or blackjack. Those games are two of the most winnable games if you know what you are doing.
Jerry. I agree. Poker because bluffing and detecting bluffs are so important a good player can win fairly consistantly. Skill can trump luck in poker. And blackjack because I believe even without card-counting, just using simpler strategies it has the best player odds of the standard casino games. The house still wins more than 50% though. I have heard that some side bets in craps are often favorable, not sure about that.
posted
Also, craps has good odds if you know how to bet. can't remember exactly, but i think it's better than the odds for blackjack.
And as for poker, i completely agree as well. But not because it has anything to do with odds. but because while getting lucky can help you in the short term in poker, poker is a skill game, and the better player you are the more money you will win.
posted
Sho'nuff, I should have said 1 or 2% is not worth the trouble to me. There are plenty of sucessful card counters out there. I would probably have such flop sweat card-counting, after a few hands the would dealer would probably signal the pit boss to watch me. I'll take your word on Baccarat, but I have doubts about video poker systems.
Posts: 6316 | Registered: Jun 2003
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posted
I also wanted to say something about the duel. The only way that #1 can survive is if he misses the first shot. #2 Then has to either hit and kill #3 or shoot and miss #1 or 3. If #2 hits #3, then its #1's shot an he can either hit or miss #2. If #1 misses, #2 must miss and the third shot for #1 is guaranteed to hit. Obvioulsy if #1 hits on the 2nd shot then he kills #2 and he survives. If #2 hits or misses # 1 or #3 on his first shot, then #3 must kill #2 and then #1 must kill #3 on his 2nd shot. These are the only ways for #1 to survive.
If #1 were to hit on his first shot, then lets say he kills #3. Well, #2 may shoot and miss once but then #1 would miss and #2 would kill him on the next shot. Obviously, if he killed #2 on the first shot, #3 would kill him next.
Btw, on the monty hall problem as well. The reason you should switch is this. When you first choose the box there is a 1/3 chance that you are right but when one is taken away your chance becomes 1/2 that you are right ONLY when you switch. If you stay with your origanal box it will always have a 1/3 chance of being right. But if you switch you have a 1/2 chance that you are right. In this case, 1/2 is better than 1/3 so you should switch.
Edit: Yeah I agree with Morbo about poker. I think poker and blackjack are skill games and while bacarat may have better odds, I can win at poker much more often.
That's one or two percent on every hand (on the mean).
So if you're playing a hand every few minutes, betting large sums of money . . . You can make a lot of money.
Posts: 15770 | Registered: Dec 2001
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quote:I'll take your word on Baccarat, but I have doubts about video poker systems.
that's fine, because i'm sure about Baccarat, and had the probabilites and the systems laid out before me. Baccarat is a pretty complex game... But about video poker, it's all heresay. i've seen some basic rules for what cards to keep in what situations, but nothing that's shown me it would win on a consistent basis.
and about card counting, i've tried it before. i practiced for a while before i went to the casino. but it's just so god damn hard to keep track of everything. especially when you're drinking, and talking to friends, and talking to random people at the table, and the cards come out so quick...I ended up just giving it up half way through. And decided to just stick with the old play by the book, with my personal preference for betting structure.
Although gambling with Icarus in AC, i have a renewed interest for perfecting the art of counting. Even though it didn't work out so hot that night.
Posts: 251 | Registered: Mar 2002
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posted
fugu, did you ever read the orginal article(can't remember if it was Wired, it might have been)? It was extremely interesting, and much more detailed. And because of it there is now a book coming out and a movie in pre-production.
and actually, the end of the article you linked is wrong. The DID get caught. Because they got greedy and ended up splitting into rival factions. some worked in groups, some worked solo. But one of the solos or side factions ended up getting caught and ratting everybody out...if i remember correctly.
whoops forget it, i was wrong. they got caught, but not by the authorities. i should've read the second to last paragraph too.
posted
Godric, that was an unsatisfying answer, IMO... Basically, the whole problem was a question of semantics, not really mathematics, it seems to me.
posted
Casinos are for people who did poorly in math.
What's interesting is that ALL the people you talk to who gamble in casinos or with bookies swear they come out ahead. Obviously the mob and the tribes run these things as a public service to distribute money to unfortunate anglos who don't understand math.
I only gamble with insurance companies and on the stock market. The stock market has much better odds than casinos, historically speaking. Lately, though...
I've never understood the appeal of gambling in a casino. I think I'd rather just run a few hundreds through the shredder and save a lot of time and effort.
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posted
You never see a poor bookie or casino owner...
Never gone to a casino myself, although it isn't necessarily a total waste of money (well, no more a waste than going to the cinema for instance). So long as you go along expecting to lose, and just use the money to have fun while you are there (and don't take any way to get any more money with you), it can be an entertaining way of spending an evening. Plus there is always the slight chance that you'll come out ahead.
BTW Richard Feynman (physicist, if any of you have herad of him) told a story about a guy he met who did make money gambling in casinos. But he didn't play the games. He'd hang around the craps table, and when some guy said he was sure he was going to roll X this time, he'd offer the guy a bet as to whether X would come up or not. Of course, he'd set the odds in his favour, but people who are convinced they are going to throw X this go usually aren't too hot on probability.
quote:What's interesting is that ALL the people you talk to who gamble in casinos or with bookies swear they come out ahead.
as one of those people, I'd have to disagree with your assumption that all these people are fibbing. And i have the books to prove it. I keep a detailed poker log which I enter data in every time I play. Including hours played and money won/lost. I can prove i'm up on the casino. granted, this isn't against the casino, but other players. but i also keep a sort of informal "gambling" log to keep track of how i'm doing when i go to casino. While i'll admit i was down for a while, a 6-8 month and approx 10 visit win streak shot me way up in the positive.
But i'll agree. The real winner is the house. Always.
That's one or two percent on every hand (on the mean).
So if you're playing a hand every few minutes, betting large sums of money . . . You can make a lot of money.
This is not entirely accurate.
(I'm a card counter, by the way, and I can say with certainty that in the long haul, that is, in the years I've been gambling, I have won more money that I have lost. Sorry if you don't believe me AK, but it's the truth. It's not even close, really. I exercise tight controls on how much I will allow myself to lose at blackjack, and unlike most gamblers, I can actually walk away when I'm ahead.)
If you wish to question my math skills . . .
Anyway, card counting can take a VERY long time to pay off. And (unless you have four MIT geeks helping you) you have to be at the table constantly, waiting for that golden moment when you have a high count. If you start at $5 for an even game (count of zero), you need to have somewhere in the ballpark of a $3,000 bankroll to have comfortably good odds of coming out ahead in the long run. (And no, I don't have that large a bankroll). And if you spend a week at the casino, you can expect to lose money two or three days, but end up ahead overall--if you gamble long enough. Ultimately, counting cards doesn't make a single player very much more than working does. In fact, unless I am simply very lucky, I might be looking at making $50 or $100 for a long night of gambling . . . when you consider I make about $18 an hour or so at my job, I pretty much get a better return on my time by simply going to work. Why do I do it then? It's fun. It engages a side of my brain that I enjoy challenging. I like the casino atmosphere. And even though the last thing in the world a counter should ever do is drink, I also like all the freebies. (And all the casino comps!)
It does not give you that one or two percent edge on each individual hand. In fact, your odds on each hand are virtually the same. What it does is enable you to predict with greater accuracy when the dealer is going to bust. So you sit around for hours, pretty much breaking even but slowly losing money, until you get a shoe that gets a high enough count of low cards late enough in the shoe for it to mean something, and then you wager frighteningly large quantities of money on the one or two hands you might have left before the shoe ends.
Greg, until you get really comfortable with it, you should not drink (buy a soda at the bar and stick a stirrer into it, so people don't know what you're drinking) and you should not talk to people. If that takes all the fun out of it, then this might just not be for you. Also, buy a casino program (Hoyle makes one) and set it to deal quickly and without chatter, so you can practice at home.
-o-
So, yes, you can turn the odds in your favor, but you by yourself cannot really make huge sums of money as a casual gambler, and it takes many hours of play to be able to guarantee a profit.
-o-
When I can't take blackjack anymore, I will play craps. The thing with craps, though, is craps is very fast. I have made wads of money in very little time, and I have also lost wads of money in very little time. Scary. Another thing I'll occasionally do is gamble on roullette progressively, which is mathematically dangerous (see "Gambler's Fallacy" above) in that if your luck is really bad, you can lose a lot of money at it. I've won this way many times (say $100 or so each time, at least 4 or 5 times I can think of), and only lost at it once (about $360). But with craps and roulette, I know that's really gambling. I'm not fooling myself about that. But hey, if some people can go to the slots, which are the worst odds in existence, then I can throw money at craps or roulette once in a while. The thing with both craps and roullette, though, is that you can lose money so much more quickly than you can accrue it.
-o-
When I lived in Miami, I used to consistently win money at Jai-Alai. In Orlando, though, I have lost money every single time I've gone. Suspicious . . .
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