I was talking with a friend the other day about probability. He stated that if you set out to flip a coin 100 times and the first 50 just coincidentally come out heads the remaining 50 will largely be tails. He’s usually a smart guy but in this instance he insisted that given an honest coin and an honest coin toss the odds before flipping influenced the out come even after actual flipping had already happened. Of coursed I dismissed this as absurd. So he demanded a test. We would flip a coin looking for series only. The first time the coin came up three ‘sides’ (either heads or tails) in a row we would bet on the outcome. He betting on it coming up breaking the series, me taking it continuing. We would bet ten dollars a series for ten series total.
He insisted that if the odds were even, as I suggested, I had nothing to worry about - I would break even. However staring at the prospect of losing $100 I balked. To my friend this validated his position. I insisted to odds would win out, he insisted real world experience is not slave to prediction.
My question: Given the above bet, are the odds still even? I question myself now.
Yes, if the coin is fair, the odds are even. (If the coin is not fair, well, all bets are off, so to speak.) You can look at it this way: You’re basically flipping the coin four times, and filtering out all possibilities except for HHHH, HHHT, TTTH, and TTTT. Each of those possibilities is equally likely (again, on assumption of a fair coin), and on two of those you win while on two your friend wins.
Perhaps one way to make it clear to your friend is by asking him if he thinks it’s possible to create trick coins by first taking an “honest” coin, and then flipping it until one randomly hits a series of, say, 100 heads, at which point one pockets it away for later nefariousness, having created, by his apparent reasoning, a coin with a strong pro-tails bias, but, presumably, with no detectable physical differences from an “honest” coin (until flipped).
Why would you take a bet on something that has 50/50 odds, especially since it’s for a reasonably large amount of money ($100)? Your chances of making money are the same as your chances of losing money (and there’s some chance that you’ll break even). I would never take a bet for that much money unless the odds were distinctly in my favor. Tell your friend to experiment with it on his own. Trying to persuade stubborn people that their ideas are wrong is generally a waste of time.
I think that you can mathematically defend your decision to decline to take part in the game. For one thing, a tossed coin is a terrible randomiser. It’s quite easy to “toss” a coin without it actually flipping over at all. For another thing, your opponent might be open to a different level of risk than you are. $100 might be a trivial amount to him, or he might be an inveterate gambler.
Well, the actual bet wasn’t for $100 at even odds, as such, but, rather, consisted of ten even bets of ten dollars. I think you’d be pretty safe taking such a bet, in this case; your chance of ending up out $100 is 1/1024, which is acceptably low-risk (to me; your feelings may vary) for the fun and/or satisfaction of the demonstration.
(Of course, do make sure the flipping isn’t rigged and you aren’t being just conned straight out of it)
I’ll just add that your friend’s point of view seems to be an aspect of the gambler’s fallacy - that probability has a memory and will ‘even out’ prior results. Nothing really to it. Partly arises from overcompensation of refuting the idea that there are ‘lucky coins’ that while perfectly fair, can be depended on to come up heads more often.
There’s also the annoying possibility that your friend ends up ahead by $30 and then insists, “See! I told you so!”
If winning dozens of dollars would be a little fun but losing dozens of dollars would make you feel crappy, then the wager need not be in your interest. I’m guessing that’s where you reluctance came from. The dollar value is not the only thing that enters a person’s evaluation of a wager.
(Similarly: Say I offer you a 50% chance of winning $1M and a 50% chance of losing $1M. $1M only goes so far these days, yet a $1M debt might break you six ways to Sunday, leaving you with no credit or house or car. This is probably not the most attactive wager.)
I’m confident he is being genuine. He wasn’t interested in demonstration exercises, he wanted me to put my money where my mouth was to prove my conviction.
I feel much as friedo does. I worried that real world experience might cut into my pocket.
Well, if he’s utterly unwilling to listen to argument or demonstration without you first proving your conviction financially, then you have two choices. You can find a level of financial commitment which is suitably high to satisfy him but suitably low to be acceptable to you. Perhaps using 20 bets of one dollar apiece, say. Or, you can just let him stew in his ignorance.
You need to set up a bet where neither side has a very good chance of winning by pure chance.
For example: You flip the coin a few thousand times. After each sequence of 3 heads in a row, you record the result on the next toss – T for tails, and H for heads. You keep going until you have 500 results recorded.
If the results are more than 60% T, then you have to pay him $100. Otherwise he has to pay you $100.
(Unfortunately, the odds are pretty good that your friend won’t agree to this bet.)
Hm. Should be possible to run, say, a fairly simple computer simulation of this, I should think. Can run the simulation as many times as you both want, for as many coin tosses per session as you both want. I should think that electrons have just as much memory as metal alloy coins.
…Right, not near as much fun as betting with real money.
Wait for three tails to come up in a row. Now your friend thinks there is a very good chance that there will be three heads next ( there is realy a 1/8 chance of this) so bet him even money. If three heads come up, he collects, otherwise you win. It’s a pretty safe bet.
He’s not offering you a fair bet. From his perspective, if he’s right you’ll lose. If he’s wrong, there’s a 50/50 chance you’ll lose anyway. He’s basically demanding that you take an even money bet when you only have a 1 in 3 chance to win.
Of course, you know that it’s really an even money bet, so you’re really getting a fair bet. But that’s not the point. If he really believes what he says, offer him this bet:
Bet that in the next 50 flips, the number of heads or tails will not deviate from average by as much as they did in the first 50. In other words, if in the first 50 flips heads came up 10 more times than tails, offer him an even money bet that in the next 50 there will be fewer than 10 more tails than there are heads, so the result will not be back to an average of 50/50. Or you can be generous and say, “hell, if you can claw back even half the difference, you win. Less than that, and I win.” Now you’ve built in a nice little percentage for yourself.
If you want to get really devious, offer him this bet: “If you think it might take longer than 50, but eventually it will come back to even, I’ll offer this bet right now - If you lose the first one, we’ll go double or nothing for another 50 flips, with the same deal. Let’s say you get back to within 6, then I win. But you can double up, and if you can get half of that amount in extra tails on the next 50, you still win. Get 3 tails more than heads in the next 50, and you win. And we’ll keep going if you like until we get to 50/50. All you have to do is get back to 50/50 at some point in the future, and you get all my money.”
I hope you can see that this is a total sucker bet, but it should be perfectly fair from his perspective.
Here’s another fun thing to do: tell him you’ve got a coin in your pocket that you flipped four times and it came up heads every time. And if you don’t want to lie, just flip a coin until you get four heads in a row, then pocket it. Now say you’ll put up a dollar, and if it comes up tails he wins, but if he comes up heads he has to pay you two bucks.
BTW, I’ve heard an equal number of gambler say that you should bet with the streak, because heads are ‘hot’, or ‘heads are running’. You’ll find roulette tables where people study the history of numbers so they can bet against them on numbers that are ‘due’, and gamblers who are studying the same numbers to determine the ‘hot’ ones and bet with the streak.
The casino loves these people, and even gives them pencils and paper to track their streaks with. Or the roulette table with have a giant electronic tabulator showing the results of the past 20 spins. It’s called fishing for the ignorant. Some guy will be walking by on his way to the bar and go, “holy cow! Six is running hot!” Then sit down and blow his beer money on innumeracy.
Of course, if there’s an inordinately long streak, there’s the possibility that the coin isn’t true, the wheel is out of balance, etc. A diminishingly small possibility (especially with the wheel, which is of course checked for that sort of thing), but still there.
But what if somebody had been flipping that coin recently and had gotten fifty tails in a row? Maybe the coin was just resetting itself when it came up with fifty heads and now it’s even again.
Or maybe the coin is really intent on defying the laws of probability and is halfway through a run of a hundred heads. Sure it’s unlikely - but when you were halfway through that run of fifty heads, how likely did it seem that you were going to get another twenty five heads in a row?
Or (and here this post is re-entering reality) you could consider that the odds of flipping fifty heads in a row are so remote that the more likely explanation is that you’ve acquired a coin that is physically unbalanced. If it comes up heads fifty times in a row, it will probably continue to come up heads indefinitely.