Probability (Food for thought)
Bean
Posts: 8,129
Here is one thing I don't understand about probability.
Take for example a coin flip.
It is stated that over the long term it will be 50% heads and 50% tails.
It is stated that even if a large number of heads comes up in a row, the next flip still has a 50/50 shot at being heads or tails.
It seems to me for the first statement to be true "50/50 heads/tails over the long run", that tails would have to have a better shot at coming up than heads after a long run of heads flips.
Take for example the unlikely occurence of 100 heads in a row. Logically I can understand that the coin has no memory of what has come before, so it MUST be a 50/50 chance. But pschologically it seems that tails MUST have a better chance of coming up.
I guess that is what makes us all gamblers ?
Bean
Take for example a coin flip.
It is stated that over the long term it will be 50% heads and 50% tails.
It is stated that even if a large number of heads comes up in a row, the next flip still has a 50/50 shot at being heads or tails.
It seems to me for the first statement to be true "50/50 heads/tails over the long run", that tails would have to have a better shot at coming up than heads after a long run of heads flips.
Take for example the unlikely occurence of 100 heads in a row. Logically I can understand that the coin has no memory of what has come before, so it MUST be a 50/50 chance. But pschologically it seems that tails MUST have a better chance of coming up.
I guess that is what makes us all gamblers ?
Bean
Comments
but would land on the felt turned over to the opposite side it started on.
Great fun from the unaware on Poker Night..
We found that flipping the coin while flat on Your thumb, and from the same side,
the 50/50 shot went out the window...
one of the best ways to "randomize" a coin toss is to stand it on edge and flick it,
that was the most random action We could count on..
Now consider the Probability of a SIX sided box(Dice)... :freaked:
http://en.wikipedia.org/wiki/Gambler%27s_fallacy
-Phil
if you toss the coin N time, the standard deviation is sqrt(N).
So over 10.000 tosses you have a standard deviation of 100. Until 3 times that is rather normal, you can go as far as 5 times and still consider it acceptable.
It means aver 10.000 tosses you can have a win/loss edge of about 300, so something in the range of 5300/4700 win/losses is ok.
In a test case with only 100 tosses you can win/loose 80/20. So the expected result is 50/50, but you can deviate from it, and loose or win a lot...
John R.
I have the probability of 1:1m of picking the winning pick 6 lottery numbers!
Statistically I have a 1:21m chance of winning.
So if I purchased 1m tickets, I beat the probability of winning, but statistically I'm still behind the bell curve.
Statistically indicates memory of what has happened in the past that could influence the future. Probability indicates the pure chance of one outcome verses another.
No matter how many times I flip that coin, I have the probability of 50/50 heads / tails, but statistically, I have a better chance of picking the right side after a number of random tosses to determine a "memory" of what has happened that could indicate future tosses.
Here's another great example that took me some to get my head around.
A single number, 28, was never picked (in 5 years of drawing) in one of the pick it lotto numbers. The probability of that number being picked never changed from 1:45. However, statistically, every drawing that number didn't get picked, it had a higher chance of being drawn.
Probabilities are drawn on a fixed (even if fluctuating) mathematical expression. Statistics are drawn on a fix set of non fluctuating mathematical expressions; things that happened in the past. Max75 brings in the curve that describes the Statistics as mathematically computed; variance/standard deviation. These calculations explain differences within a probability set, and link that into a statistical set.
In a fixed expression like the coin toss, or a particular card game, the probabilities never change. They are fixed because the variables are fixed. In the same expression, statistics change from one sample to the next sample, because the statistics are based on previous samples. As far as a card game goes, the probabilities change when you add additional players, but the ratio of probabilities is still a fixed set for a given number of players / decks used.
In stats, our team used 10 coins, each dropped from edge onto a varying speed spinning table top with random "wedges" on it. Each coin's outcome was tracked as a single sample, and all 10 as a percentage was tracked as a sample, 10,000 samples (drops). This provided 11 samples, 10 of which were singles, one as an average, that when applied to the standard deviation, brought all single samples within an 50/50 average (thus the probability ratio of 50/50), but the averaged sample returned a 60/40 ratio. It was hard for our team to figure out why the averaged sample rate was so different then the single rate. It turns out that an averaged number is a measurement of standard deviation. Once we came to that conclusion, and removed the standard deviation calculations from our base computations, the average sample hit smack dead at 50/50.
KK
The problem, Bean, is that each flip is not related to the previous or subsequent flips. There is no information stored in the coin or environment that would bind the flips together to make your statement work, so they must be considered unique to each other. Therefore, the odds don't accumulate over time... and since there are only two sides, each toss has a 50% change of landing on either side.
Bill
About 15 years ago I was playing 25 cent Roulette at the Palace Casino in Biloxi, MS. (The casino is still there, although Katrina washed their boat away the high rise ground complex survived.) This dude who was being called to get on his tour bus to leave walks up and plonks a quarter on RED. Wheel comes up black. Bye-bye, 25 cents. Dude plonks down 50 cents.
This pattern of doubling up to recover your losses is called a Martingale. It's called that because the rather racist 18th century Frenchmen who coined the term thought the people of Martingale were extremely stupid.
So the dude loses the 50 cents and plonks down a dollar. By the time he's up to eight bucks I started betting against him on general principles. My companion was betting a wheel sector that happened to contain 17 black and she was cleaning up. He kept doubling until he hit the wheel limit of $1,000, tossed in the rest of his cash (which wouldn't have covered his losses if it hit), lost that, then with no more money to bet watched as the very next spin came up red.
No matter how unlikely it seems it can actually happen, and will if you hang around the casino long enough. I was hanging around the casino quite a bit in those days and on four or five different occasions I've seen the entire Roulette history scoreboard, which displays the last 20 spins, all lit up in one color.
The reason it eventually evens out isn't that there is a bias pushing back after 20 red spins; it's that, if the wheel is honest, at some point in the future it's just as likely to start a run of 20 black in a row. But there's no guarantee of which point in the future when that will happen, only that the longer you wait the more likely you are (but never at all certain) to see it.
Yep.
That story would be just perfect if you could have ended it with the guy finding he's missed the tour bus...
The answer is here, but see if you can figure it out for yourself before peeking.
-Phil
The method that you describe on the Roulette table is exactly why there is a "maximum betting limit". Theoretically if you had an endless pool of money that you could just keep doubling and at some point in the future when the wheel does favor the other direction your net gain, even though you might have a nice crowd watching you, will only be the initial amount you placed before you started doubling it. There are many other casino games that this 'loop hole' transcends into and thus the reason for a maximum betting limit. It works sometimes, I myself have turned $300 to $3000 on the Roulette table this way, but I don't recommend it.
Beau, if you managed to 10x your investment with a Martingale strategy you are one of the luckiest people around -- and I mean that in a bad way, because early gambling luck is one of the unluckiest things that can happen to you. For us the Roulette experiment was a little side trip on a mostly profitable run of small scale advantage play -- playing tournaments and milking giveaways, coupons, and small comps to turn entertainment expense into a small entertainment profit. Unfortunately, most of what we were doing then isn't possible now because the industry has consolidated and gotten tight and they just don't give stuff away like that any more.
Then our close friend whom I called X in a rather prominent web article about what happened started card counting successfully, and got my wife involved. That paid off our house. Too bad our friend lost the half million or so he won at blackjack in the stock market. He lost most of it on September 12, 2001 thanks to trading heavily on margin.
I only gamble very occasionally now. There's nothing like seeing your friends rake in a couple of million bucks (as a team) to sour you on letting the casino do that to you even for five bucks at a time. Last time was our trip to Montreal a couple of years ago. We checked out the casino in the arty World's Fair building and I had a surprisingly lucky run at the craps table.
When gambling recreationally, I only play Craps or Mini-Baccarat now, so I can avail myself of the free alcohol without making the house edge even worse by playing stupid.
Another life ago I used to play professional pool, which lasted for about 10 years. Early on when I started playing, our team qualified for Vegas. ESPN 9-ball tournament. Anyway, back then a few of us decided to make an annual trip to Vegas whether we qualified or not. In order to survive out there you MUST be strict with yourself. Out of the ten years that I went out there I would either double what I took with me, or come out even. ...and to me coming out even is still ahead, it means that I at least came back with an amount as though I had never taken the trip in the first place.
My game was the Roulette table and I exclusively played on that. But I didn't mess with the 2:1 odds (i.e. Black/Red, Odd/Even, High/Low), I played the 3:1 odds and applied a Martingale technique. This worked for me, but like I said I wouldn't recommend it. It's a rough gig.
On my Vegas visits, I would take $2000 for 5 nights. That breaks down to $400 a day which includes the Hotel stay and food. Each day I would have $200 to $300 to gamble with. If I lost it all, I was done for that day, the next day was a new day and again I had $200 to $300. If on one of those days I got lucky and won something, the next day I would NOT dip back in to my previous winnings and I would once again have $200 to $300. That method at least guaranteed food and a place to put my head. Air fare at the time was about $250.
Lucky? ...perhaps, at the time I thought otherwise, but whatever I did made Vegas nervous. I was "spotted" so many times with the wandering overhead camera. Sometimes you can catch a glimpse of a little spotlight about the diameter of a half dollar... Then you had the men in black suits standing behind you looking over your shoulder with the 'pig-tails' coming out of there ears... Yup! ... I was just young and dumb and filling an interesting moment in my life.
If you see it as entertainment expense and limit your losses, you have won.
I have seen many unfortunate gambling stories. In the early days with the comp schmoozing and tournament play it was almost all fun; you see the same people, they're always there and they're not on a degenerate curve either, it was like a little bi or tri-weekly get together.
At the high roller tables, not so much.
Another true story: At the height of the card counting team, right before my wife quit, she spied what looked like a good opportunity and sat down at a table at the Hard Rock in LV. She proceeded to count cards and spread her bets and was amazed that there seemed to be no attention on her, so she spread her bets more aggressively to make more money (but exposing her card counting strategy more) and still no attention, at that point in time an amazing thing. She got lots of action and played for several hours before just plain getting tired and leaving. She noted that there seemed to be lots of people standing around watching them but hey, they were high rolling so whatever.
When she got back to her hotel room there was a new message on the web forum they frequented: "Just saw Y playing with Brad Pitt at the Hard Rock."
She played blackjack for several hours with Brad Pitt and didn't even realize who he was.
PS on realizing I didn't mention this -- the opportunity was that it was a high limit table and he was the only other player.
Then, when gambling, invest one of the smaller amounts. If you end up with more than that amount, put it into the other pocket, not to be touched for that day.
Entertainment then costs what is in your left pocket, with your "winnings" being what ends up in the right!
Exception to this is poker. That's a skill game, and requires a very different strategy. (Mrs and I are fairly good poker players, she is ranked --and dangerous enough to get free rooms, just by announcing she will be in town!)
But if you look at random walk theory, some rather disturbing behavior exists where you get long runs of remainding in the money or out of the money.
As best as I can see, probabilty is NOT reality. (The house in Las Vegas will always cut you off if you win too much, regardless of having probabilty in their favor.) Recently, someone mentioned a device that had 228 years of Mean Time Between Failure. It seemed quite absurd to me as no computer hardware has been in existence for half that long and the product was only guaranteed for 3 years.
If I have a 1% chance of survival, I will still have lunch and hope for the best. If 90% of the students will fail an exam, I will still try to pass it. I am NOT a statistic - neither is anyone else.
I've been to Las Vegas. After loosing $20 in 20 minutes, I left.
Also, just because something has an X% probability of occurring does not necessarily mean that its behavior is driven from first principles by chance. It may simply mean that there are variables not known to the observer which, if they were known, could completely predict the outcome.
-Phil
for distribution based on coin flipping and determine
the value of pi.
Mathematics, statistics, and probability are somewhat useful abstractions. And I guess, so is language.
Yes, I use that to advantage. When there have been multiple coin flips and all the same, I will guess the opposite. Most of the time I win in that situation.
Then when playing poker with say 2 other people, if I have not won for several games, I will bet more heavy as I am more likely to win (not real money). And if another person has won two games in a row, very unlikely he will win again, so if it gets down to between me and him, I'll bet heavy. And I frequently win when it is getting to be "my turn" to win.
BUT, people will sometimes win 3 times in a row, but quite rare.
As to gambling, I NEVER bet real money as I realize it is a guess at best and never a sure thing ever. That coin might come up the same again or I might lose at poker even though it is beyond my turn to win.
I've invested my money in energy saving things and that has turned out to be a "SURE BET"! Every month I get a lower electric bill and will continue to do so for the rest of my life... Can't beat those odds!
No, it only seems like you do because of selective memory. This has really been studied exhaustively, not just from a "pure math" angle but with real people making real guesses. In games like coin flips, Roulette, and Craps, the game has no memory, but you are more likely to remember a run and its aftermath than a more chaotic series of results.
Your statement here is the basis for the inevitably disastrous Martingale, all those people keeping scorecards at the Roulette wheel and Baccarat table, and just about every Craps betting system ever invented. And none of them work. If they did, the casino wouldn't let people use such systems. (Just try keeping a scorecard at a blackjack table, where the game does have a memory. You'll last about two hands.)
This is an important thing. The implications of randomness are subtle and our intuition is defective. The casino industry exists because of this. If you actually record thousands and thousands of coin flips, dice tosses, or Roulette spins -- and people have actually done this -- you will see that the math is right and intuition is wrong. But you do need thousands of samples to demonstrate the truth of it; in shorter runs you will see outliers. In fact, if you don't see outliers in a short run it suggests something is wrong. If you're faking the results, you wouldn't be "dumb" enough to serve up heads ten times in a row, would you? But an actual coin will, and if I play your game long enough and never see such a run it's one of the surest clues available that the game is rigged.
Anyway, you do give one excellent bit of advice: If you want to use a system like this then by all means don't gamble with real money.
And, since we're on the topic and the mods don't seem to mind, I suppose there's no harm in posting my own memoirs of that previous life when I spent almost as much time at the casino as I did at work:
A Casino Odyssey: http://www.kuro5hin.org/story/2001/7/19/181127/355
As to craps, I found that if I simultaneously bet on the come and don't come with the same amount of money, then placed odds on don't come, then I would outlast most people on the table. I would continue to have money to play with while everyone else lost their money and had left the table.
Also some casinos had a FIT when I would do this. And if they don't like what you are doing or say it is "bad luck", etc., than that of course is the thing to do. :smilewinkgrin:
Also I found that if I played the "win a zillion dollars" slot machines... I would play each machine until I won something. A quarter or 50 cents. Then I would go to the next machine and play until that won something. And do this with all the machines in each casino. (Each casino would have about 10 of these machines.) Then I would go to the next casino and play the same machines, then the next, and next. By the time I got to the end of the strip, I would have usually won about $10.
Then there was my "walk into the casino and listen" system. If you walk into a casino and hear a lot of coins dropping from the machines, then that is a good casino to play slots in. If you hear dead silence (no one winning anything), then don't waste your time there.
And the Indian casinos I have been to are dead silent.
So the above two things are ways to break even or make a little money, but they both take a very long time and a lot of work with little or no returns. But the slot thing is a good way to get some exercise if you want to walk off that buffet lunch!
The general thing though is to look at the casinos. Look at all the fancy buildings and furnishings. How do they pay for that? Well you pay for that!
But even Vegas can't win all the time. Last I read, they were having financial problems due to the downturn in the economy.
Of course, the optimal strategy may not be the most fun if you really enjoy sitting in front of a slot machine for hours and don't mind going home broke and smelling of stale cigarette smoke.
-Phil
What are the chances a regular person can continue to flip a coin repeatedly and have it come up heads again and again?
Look at it from an engineering perspective. The human body is not a perfect machine. There will be differences in the flipping process. A bit more speed. Slightly different angle of toss. Maybe start out sometimes with heads up in the hand, other times tails up in the hand. Etc.
Sometimes, when Mrs is playing a longer poker tourney, I will go through and examine the payouts of all the machines. Some have rather liberal small amount payouts, others linear, others "hockey stick" type structures.
For a given bankroll investment, there are smarter wagers than others. If the cost of the odds of winning is less than the payout, that's generally a good wager. If the inverse is true, then it's generally not.
Weighing what it will pay, the size of the bet, and the payout, often reveals sweet bets that are not max bets. (which are always the smart wager, but quite expensive, and they make it that way on purpose to leverage what Phil mentioned above "gamblers ruin")
On those machines, a series of modest bets can pay back fairly significant amounts, significantly lowering the cost of smart wagers. If I'm in the mood for some play on simple slots, that's what I'll do. Get the free drink, then walk around checking out the pay structures, doing a little math, then placing the inexpensive bets, hoping for the payout.
Just betting smaller amounts on the slots isn't wise, unless that consideration has been done. Where it makes sense, some very nice payouts are to be had for modest amounts of money. Those are fun, and benign as the amounts of money involved are easily managed, but one has to do the work...
A lot of the risk on slots is failure to match the bet to the risk profile of the machine. They all are very risky, that's the point, but playing the smarter wagers will decrease the house advantage considerably. IMHO, this is a big part of why there are so many betting options available. Where people can choose they will! (even when the correct choice is obvious, they will still choose poorly) That equates to more revenue over time for the house. It always remains possible to determine the smarter wagers, but if they confuse the matter, and add booze, not probable, which is the revenue increase for them, right there. The product of this is most players actually play at a risk level considerably higher than is presented.
I like the Craps strategy mentioned! Nice one. That factors out a lot of the variance in the game. Nicely done. At some point in the future, I may give that a go. (I'm there in that environment once or twice per year for conventions and such)
Re: flips, again and again. each flip is 50/50 However, if combinations are the subject of a wager, that's a state, and the odds then change considerably. The key is to link the wager to a state, not see a state internally, yet have the wager be linked to discrete events...
This is basically not playing except for the rent you pay whenever there's a 12 on the come-out roll and you lose your pass bet but the don't pushes. The don't odds bets are technically called lays, because you get paid less than 1:1 for them, e.g. you win a $20 lay on the 4 you get paid $10. You win more often but when the bad streak comes, which it will, it wipes you out.
Yes they are, mainly because people like you don't have enough cash on hand to go lose to them.
A friend and I looked into the probability maths in more detail, covering your losses. Say the odds are stacked slightly in the casino's favour. You gamble, and you lose. But generally it is not because of the odds stacked against you - it is because you run out of money first.
So we did some maths looking at what would happen if the odds were exactly 50:50. You still lose, because they are bigger. We then looked at what would happen if the odds were actually stacked in your favour. You still lose, because they are bigger, and because you get greedy and want to win more. And you will gamble even more if the odds are in your favour - surely you can't lose, right? We plugged in some numbers - say the casino has 1000x more money than you, and what odds would you need to eventually win using a 'cover your losses' scheme. It is surprising how much the odds can be hugely in your favour before you guaranteed of winning. And even with a not 50%, but 90% chance of winning, nothing is certain, because along can come that run of 100 heads in a row and if your pockets are not deep enough...
So the way to beat the casino is to be bigger than them!
But I also know from experience this: The math works. Intuition is wrong about these things. If you do what the math says and the math says you can win, you will win; if you don't do what the math says you will lose. I spent about eight years living the truth of that and even though I decided it was more fun (and more reliable) to make things that work for people, I did learn this particular lesson better than most people ever do and I'll never forget it.