Probability Question for Dice

I have this argument with my father often, but don't remember enough stats to answer the question correctly. This is regarding craps, but it is more easily illustrated with a coin toss. Let's say we're flipping a coin which we know is 50/50. Probability says we should get 5 heads in a row about 3% of the time. If we've gone a very long time without seeing 5 heads in a row, my father believes there is a greater and greater probability of seeing those 5 heads, and so you can start betting more on that outcome. I know what the basic math says, and that every throw is independent of another, but how do you explain on the one hand the probability is always 3%, but if you haven't seen the 5 heads come up in many throws, that you shouldn't have any greater expectation that they will show up. It seems like a paradox but I'm sure it is because I'm missing something about how the statistics are interpreted, or variance or something.

Any help????

Dan

Quote: Bushka

Let's say we're flipping a coin which we know is 50/50.
Probability says we should get 5 heads in a row about 3% of the time.

Probability needs to know the number of trials.
The number of trials also determines the probability of a run of any length

in 10 flips a run of length 5 is 10.9375% not 3%

start here.
good links to the math
http://www.pulcinientertainment.com/info/Streak-Calculator-enter.html

It also shows a link to a WoV thread on doing the math yourself instead of using a streak calculator


Coin Flips are independent events where a series of the same coin flips (streaks or runs) are dependent events.
You can not have 2 heads in a row without the first flip being a head.
Conditional probability is now in force.

The average number of flips to see a run of 5 heads in a row, for example, is 62
https://wizardofvegas.com/forum/questions-and-answers/math/8141-on-average-how-many-trials-will-it-take-to-see-a-streak-of-8-qs-for-fun/

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Yuor father is letting the emotion of things needing to balance out get the better of him. Yeah, it will eventually balance out, but that doesn't necessarily mean it's gonna happen anytime soon.

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Past performance is no indicator of future outcomes. This same phrase (or one like it) is seen on all investment advertising. Gambling is no different than investing in that regard.

Unless you feel that past performance is an indicator of an unfair coin / loaded dice / bias wheel, etc, then the next outcome has exactly the same probabilities as the first outcome.

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Quote: Bushka

If we've gone a very long time without seeing 5 heads in a row, my father believes there is a greater and greater probability of seeing those 5 heads, and so you can start betting more on that outcome.

Your father is right.
The more trials the greater the chance of seeing at least one run of length 5.

Your 3% value is only for 5 flips of a coin and then you start over for the next 5 flips.

If after the first 5 flips no run of 5 has happened keep flipping, the probability increases.
In 50 flips
0.551875322953617 chance of at least 1 run of 5
100 flips
0.810109599196358 chance 0f at least 1 run of 5
300 flips
0.993877629387255

No Gambler's Fallacy here, just more trials

Quote: sodawater

Guido, I think you're misinterpreting the question. It's pretty clear to me that OP is talking about betting that the next 5 flips will be 5 heads -- which is always 1/32 no matter what happened on the prior tosses.

I think you did not understand the OP question. I do not see anywhere about the very next 5 flips.
He keeps flipping until the event happens.

So we now disagree and are both correct.

You for any 5 flips, me for any number of flips.

Quote: guido111

I think you did not understand the OP question. I do not see anywhere about the very next 5 flips.
He keeps flipping until the event happens.

So we now disagree and are both correct.

You for any 5 flips, me for any number of flips.

He's talking about betting on it. Nobody will let you book a bet post facto...

Quote: Bushka

my father believes there is a greater and greater probability of seeing those 5 heads, and so you can start betting more on that outcome.

Any help????

Dan

OK. I still say the gambler can bet for a run of 5 for more than just 5 flips.
Say for the next 10. Nice parlay if he hits it and still has a 10.9% chance of success not 1/32

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The non-intuitive nature of gambler's fallacy is the foundation for which Las Vegas is built upon.

There is nothing to be ashamed of in terms of believing in the past events affecting the future ones. Anyone with this belief has many others who also believe this.

But it is also not the _only_ false belief upon which money is drained from the pockets of fools believing untruths and testing their belief systems with their pocketbooks.

Some get lucky testing their belief systems in fact, but that does not mean that the belief was true.

The human brain is absolutely exploitable.

Yet Las Vegas is having it's own troubles with it's false beliefs.

Such is the case with the boom economy and boom real estate of years past.

So at least there is some rectification of hubris on the opposing side.

Time will tell what is next, but the human belief system is a powerful thing, even when it holds false beliefs, some people with false beliefs have a tremendous amount of power.

Quote: sodawater

OP even gets the probability for the next 5 flips being heads correctly. That's what he was talking about. It's a simple gambler's fallacy question. Has nothing to do with getting 5 consecutive flips in N tries.

Quote: Bushka

how do you explain on the one hand the probability is always 3%, but if you haven't seen the 5 heads come up in many throws, that you shouldn't have any greater expectation that they will show up.

Simple answer: The probability of 5 throws all be heads is 3% (more precisely 1/32).

Just because you didn't see any 5 heads in many throws doesn't mean they are any more "due" than 3%.

Quote: Bushka

I have this argument with my father often, but don't remember enough stats to answer the question correctly.

This is regarding craps, but it is more easily illustrated with a coin toss. ..........................
Any help????

Dan

Yeah, IF This is regarding craps, what IS the Craps question?
I hate flipping coins.
Screw the coin toss.

What is it... Someone wants to make a craps bet for 5 in a row, after seeing no five in a row for what???
18 in a row? A different trigger?

Again, what IS the Craps question?
and why not ask a craps question in the proper forum?

I do not watch too much South Park.
I know who Laura Nyro was, and even like some of her music, not all, some.
Stacy F might really like her.

Thanks for all the responses! It really is more of a math question so I shouldn't have mentioned craps, but I may post over there at some point. I've read all the replies but I'm still not satisfied that I have a clear understanding on how to change someone's mind about this (or that I understand what I need to know). Some of you (guido) are saying that the longer you wait (ie more and more flips) the greater chance you will have 5 heads in a row. Others are saying it's 1/32 no matter what, and others are saying it will even out but you can't tell when. I suspect you are all correct, but a clear answer isn't jumping out at me. Even the Gambler's fallacy Wikipedia didn't directly address what I'm looking for.

Let's say we've been flipping coins and Guido's calculator says we have an 88% chance of seeing at least one string of 5 heads in the first hundred tosses. We've flipped 80 times so far and still no string of 5 heads. Now we say, "Geez let's start betting on the 5 heads because there are only 20 flips left and we have an 88% chance in this first 100 flips." I know the probability of it happening in the next 20 flips is the same as in the 1st, 2nd, 3rd or 4th group of 20 flips, but how do you square that against the fact that 5 heads will show up a certain percentage of the time over the long term, and at some point if they don't show up, they will be grossly under represented in the completed throws? If they are grossly under represented, why can't you say the probability of the 5 heads coming up increases, and bet on that?

I guess the answer is actually more a matter of logic rather than mathematics, and I think that's what I'm looking for first. The math can confirm the logic, but it won't convince anybody who thinks otherwise.

Thanks again!

Quote: Bushka

Thanks for all the responses! It really is more of a math question so I shouldn't have mentioned craps, but I may post over there at some point. I've read all the replies but I'm still not satisfied that I have a clear understanding on how to change someone's mind about this (or that I understand what I need to know). Some of you (guido) are saying that the longer you wait (ie more and more flips) the greater chance you will have 5 heads in a row. Others are saying it's 1/32 no matter what, and others are saying it will even out but you can't tell when. I suspect you are all correct, but a clear answer isn't jumping out at me. Even the Gambler's fallacy Wikipedia didn't directly address what I'm looking for.

Let's say we've been flipping coins and Guido's calculator says we have an 88% chance of seeing at least one string of 5 heads in the first hundred tosses. We've flipped 80 times so far and still no string of 5 heads. Now we say, "Geez let's start betting on the 5 heads because there are only 20 flips left and we have an 88% chance in this first 100 flips." I know the probability of it happening in the next 20 flips is the same as in the 1st, 2nd, 3rd or 4th group of 20 flips, but how do you square that against the fact that 5 heads will show up a certain percentage of the time over the long term, and at some point if they don't show up, they will be grossly under represented in the completed throws? If they are grossly under represented, why can't you say the probability of the 5 heads coming up increases, and bet on that?

I guess the answer is actually more a matter of logic rather than mathematics, and I think that's what I'm looking for first. The math can confirm the logic, but it won't convince anybody who thinks otherwise.

Thanks again!

I think you're having a problem with the non-intuitive nature of gambler's fallacy.

One way to confirm that it doesn't matter what just happened is to perform actual trials in the real world with coin tosses.

Wait until you go more than 100 tosses without five in a row and start seeing how long AFTER that before you get five in a row.

You will see that the average amount of time to get five in a row after 100 tosses without five in a row .. with enough samples .. is .. the .. same.

I think the only real question here is why does the human brain think that the history of what has happened with random outcomes affect future random outcomes?

The short answer is "because they are random" and that by definition, random numbers are independent of previous outcomes.

A coin-toss and a dice throw, is considered to be random. But true randomness is not defined by a coin-toss or a dice toss.

True randomness is a theoretical concept, that, by definition, has an outcome that doesn't not depend on what already happened.

I hope this helps.

Quote: Ahigh

I think the only real question here is why does the human brain think that the history of what has happened with random outcomes affect future random outcomes?

I think what you are saying is this (correct me if I'm wrong): If there were even 100,000 tosses with no 5 heads coming up, it wouldn't have any bearing on the next 100,000 tosses. You would expect the next 100,000 tosses to show the expected number of 5 head strings, and you would accept that there was an extraordinarily flukey event in the first 100,000 tosses. If you combined the two trial data to make 200,000, you would just accept that the 5 heads were under represented by a fluke. You would not expect the second group of 100,000 to have double the expected number in order for the 200,000 to look mathematically inline with expectations.

I'm thinking it is that last sentence that is the key to the explanation I'm looking for. The outcomes are always going to approach their mathematical percentages over the long term (law of large numbers) but nowhere along the line do you have to have one set of 100,000 flips counteract another set of 100,000. In fact, you could flip 10 million times, approach the theoretical numbers asymptotically, and never have one subset of 100,000 contain more than the expected 5 head strings.

I think this answers Ahigh's question above. People believe that in order to reach a mathematical expectation, there has to be a flood of expected results somewhere along the line if that result is, to this point, under represented. However, the theoretical result can be reached from the "under represented" side or the "over represented" side or by "flipping back and forth" but there is no preference mathematically for one or the other.

Do I have that right?

I think you've got what I was trying to say. There are more mathematical guys around here. But if you detach dice and coin-tosses from the definition of random, and just consider those "humanly acceptable approximations to random outcomes" this may or may not help.

The concept of randomness, in my belief system, is 100% theoretical. Just like the concept of a circle. Or the concept of a balanced die (that's another story).

100% theoretical concepts don't even truly exist. But you can in fact define their properties and use approximations of those theoretical concepts quite readily without any normal person ever being able to distinguish between the two.

Quote: Bushka

I think what you are saying is this (correct me if I'm wrong): If there were even 100,000 tosses with no 5 heads coming up, it wouldn't have any bearing on the next 100,000 tosses. You would expect the next 100,000 tosses to show the expected number of 5 head strings, and you would accept that there was an extraordinarily flukey event in the first 100,000 tosses. If you combined the two trial data to make 200,000, you would just accept that the 5 heads were under represented by a fluke. You would not expect the second group of 100,000 to have double the expected number in order for the 200,000 to look mathematically inline with expectations.

I'm thinking it is that last sentence that is the key to the explanation I'm looking for. The outcomes are always going to approach their mathematical percentages over the long term (law of large numbers) but nowhere along the line do you have to have one set of 100,000 flips counteract another set of 100,000. In fact, you could flip 10 million times, approach the theoretical numbers asymptotically, and never have one subset of 100,000 contain more than the expected 5 head strings.

I think this answers Ahigh's question above. People believe that in order to reach a mathematical expectation, there has to be a flood of expected results somewhere along the line if that result is, to this point, under represented. However, the theoretical result can be reached from the "under represented" side or the "over represented" side or by "flipping back and forth" but there is no preference mathematically for one or the other.

Do I have that right?

Yes. Moreover, the law of large numbers refers to the ratios, not the absolute numbers. The ratio of heads to tails will approach 50/50, but the actual difference between heads and tails does not decrease (as your father might think) but actually increase over time. It is more likely that abs(heads - tails) is greater the more flips you do. To test this: flip a coin 10 times, then 100. In which trial was the ratio closer to 50/50? In which trial was the difference between heads and tails closest to zero?

Quote: Ahigh

100% theoretical concepts don't even truly exist. But you can in fact define their properties and use approximations of those theoretical concepts quite readily without any normal person ever being able to distinguish between the two.

I think I veered into the Zen forum. :) I understand what you are saying. I hadn't considered that dice are a slightly imperfect way of illustrating randomness....hmmm.

Quote: MathExtremist

Yes. Moreover, the law of large numbers refers to the ratios, not the absolute numbers. The ratio of heads to tails will approach 50/50, but the actual difference between heads and tails does not decrease (as your father might think) but actually increase over time. It is more likely that abs(heads - tails) is greater the more flips you do. To test this: flip a coin 10 times, then 100. In which trial was the ratio closer to 50/50? In which trial was the difference between heads and tails closest to zero?

Yes, I understand. I recall reading something in a blackjack book I had years ago. As a refinement to what I was saying above about the 100,000 flip trials: I think the book said that there was actually a greater probability that you would approach 50/50 over millions of flips from either one side (heads) or the other (tails). For instance, it was not as likely that the first million flips would favor heads, and then say the first 3 million would then favor tails, the first 6 million favoring heads again, and so on. There could well be a flip flop from one 100,000 group to the next, but the cumulative total will tend more to favor one side or the other. I may not have that exactly right, but it is something like that. I thought someone else might know the precise answer.

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Quote: Bushka

Yes, I understand. I recall reading something in a blackjack book I had years ago. As a refinement to what I was saying above about the 100,000 flip trials: I think the book said that there was actually a greater probability that you would approach 50/50 over millions of flips from either one side (heads) or the other (tails). For instance, it was not as likely that the first million flips would favor heads, and then say the first 3 million would then favor tails, the first 6 million favoring heads again, and so on. There could well be a flip flop from one 100,000 group to the next, but the cumulative total will tend more to favor one side or the other. I may not have that exactly right, but it is something like that. I thought someone else might know the precise answer.

ME basically spelled it out.
As the number of trials increases the percentages or ratios approach 50/50 but not the actual values, they are more likely to get further away from 0 (the difference between H and T).
This is also the difference between the Relative frequency and the Absolute frequency


A few demonstrations of the law of large numbers where the ratios converge but not necessarily the absolute values when p=0.50.
I like this one
http://www.tinafad.com/coin.php

http://demonstrations.wolfram.com/LawOfLargeNumbersComparingRelativeVersusAbsoluteFrequencyOfC/



The idea of one side staying in the lead way more than the other over many trials,
that is due to the arc-sine law of a simple random walk.
Many text books talk about this.

Stock traders would be well served to learn more about this principle.

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Quote: 7craps

A few demonstrations of the law of large numbers where the ratios converge but not necessarily the absolute values when p=0.50.
I like this one

Very cool. Thanks!

Quote: sodawater

Another cool thing about the law of large numbers is the probability of getting X heads in X flips.

For example, the chances of getting 6 or more heads in 10 flips are 37.7 percent.

But the chances of getting 60,000 or more heads in 100,000 flips are just 0. There is no way to reasonably express that number except 0.

Makes a lot of sense. I feel much better now. :) I think the key phrase for me to use is that if you haven't seen any result you've been expecting (like 5 heads in a row, or a hardways 10 or whatever), you can expect to see more of them in the following rolls, but no more than probability tells you. There's no "catching up" with results greater than probability determines.

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Yep, the CLT
Just using simple math.

100,000 flips
EV = 100,000 * 50% = 50,000
Variance is N*P*(1-P) = 25,000
Standard Deviation = 158.1 (just the square root of variance)
+-4SD (8sigma)

50,000 +/- 632.4

1,000,000 flips
EV = 1,000,000 * 50% = 500,000
Variance is N*P*(1-P) = 250,000
Standard Deviation = 500 (just the square root of variance)
+-4SD (8sigma)

500,000 +/- 2000

as the range increases the relative frequency does not

When somebody comes up with a belief like your father's, I usually respond with something like this:

The probability of having 20 consecutive roulette wheel spins (on a double-zero wheel) all being red is 1 / 1,762,869.

However, the probability of having 19 consecutive spins all being red, and then the 20th spin be black, is also 1 / 1,762,869.

In fact, in the coin-tossing case that you mention, I think it's more likely that the coin is not fair, and you should consider betting more that five heads in a row wouldn't happen.

Quote: ThatDonGuy

When somebody comes up with a belief like your father's, I usually respond with something like this:

The probability of having 20 consecutive roulette wheel spins (on a double-zero wheel) all being red is 1 / 1,762,869.

However, the probability of having 19 consecutive spins all being red, and then the 20th spin be black, is also 1 / 1,762,869.

In fact, in the coin-tossing case that you mention, I think it's more likely that the coin is not fair, and you should consider betting more that five heads in a row wouldn't happen.

I've tried that and it doesn't sink in. However, I sent him a long, detailed email last night on why, for instance, if you have a bunch of reds on roulette, that the probability gods don't have to hand out an overload of blacks to balance the reds out. He understands it completely, or nearly so. I think it has to sink in for a few days until somebody is able to take that information and apply it to whatever system they think will work.

Thanks.

What people, especially chasing gamblers, fail to realize is that what seems like an extraordinary run of statistical oddity isn't as strange as they choose to believe.

Lets say someone is betting coin flips, and he's betting heads. Right out of the box, 16 out of 20 come tails. He figures, this can't last forever, so he doubles his bets on heads. 13 of the next 20 come tails. He bets even more. Has a run of 10 each heads and tails in the next 20 before seeing 13 of the next 20 after that come tails. He goes broke, thinking what he just witnessed was as rare as a blue lobster. However, what just occurred was a run of 28 heads out of 80 flips. The chances of that occurring are around 205-1, which is similar to the chances of being dealt pocket aces in a hold 'em game. Now, for those of us who play cards, we're not totally shocked when we get dealt aces because we get them all the time. Every so often we even get them dealt a couple of times in a few hands.

When you're dealing with 100 or 200 events, they are the same ratio to a pool of 100,000 as being dealt 2 cards in a nights worth of poker, or an hour of online poker. It's just not something that will neccessarily balance itself out in the next 100, 200, or even 500 events. It probably will, but variance is a funny thing, and there are a lot of broke gamblers living in boxes who found that out.

shakhtar - that's an interesting way to put it into perspective.