The next step requires you to look at the cases where there are either 1, 2, or 3 heads in the first 3 flips. If there's 1 head you just look at whether it's followed by a head. If there's 2 heads you flip a coin to decide which one to pick then look at whether it's followed by a heads. If there's 3 heads you randomly pick one of them and look at whether it's followed by a heads. In other words, you randomly choose one of the heads in the first 3 flips, check it's successor for heads and let that result represent the result for the quartet of flips. So there's one result for each qualifying quartet of flips.

By this method

HHHH counts as 1 successful hothand even though there are 3
THHH counts as 1 successful hothand even though there are 2
while
TTHT counts as 1 failed hothand when only 1 candidate failed

The probability space for this step requires you look at all 14 possible ways of getting at least 1 head in the first 3 flips, ie. all possible ways of getting a qualifying quartet of flips. Some qualifying quartets will have a probable result rather than determined as above. For example HTHH is 50-50.


PairTheBoard

So are these examples of possible outcomes and associated probabilities:

P(HHHH, do nothing) = 1/16, hothand;
P(THHH, do nothing) = 1/16, hothand;
P(HTHH, make random choice between 1st and 3rd flip, 1st flip chosen) = 1/32, not hothand;
P(HTHH, make random choice between 1st and 3rd flip, 3rd flip chosen) = 1/32, hothand;
P(TTTT, do nothing) = 1/16, not hothand?

Right, except you've already conditioned on "at least one head in first 3 flips" so TTTT and TTTH are excluded and the denominator going into each of the 14 equally likely qualifying cases is 14 rather than 16.


PairTheBoard

I'd prefer not to look at it like that. Four coins are flipped and then something else happens. TTTT and then do nothing is a possible outcome. I'm aware that P(A given B) is being considered, where B is proper subset of the set X of all outcomes, but that seems standard, since P(A given X) = P(A). So it makes more sense to me to have the denominators as (multiples of) 16.

Yea, you can do it that way. You then have three possible outcomes of the experiment. Hot Hand Success, Hot Hand Failure, and N/A with P(N/A) = 2/16. You compute the P(Success) for the 14 qualifying cases as some multiple of 1/16 for each (as you were doing). You add those up (they are mutually exclusive) to get s. You then condition on the event E=(Not N/A) which will give you the conditional probability,

P(Success Given E) = 16s/14 which puts 14 in the denominator at the end.


PairTheBoard

Twenty-odd outcomes. You have listed three events.

Not quibbling, just reality-checking.

Say you repeatedly flip n coins, count the heads, and chart the results. You will notice they fall on a normal distribution, or bell curve.

http://probability.ca/jeff/writing/margerror.html

Looking at the first 3 charts, you can see that the distribution is getting more and more narrow as you go from 10 to 100 to 1000 coins, but the probability of getting exactly 50% heads goes from 25% to ~8% to 2.5% respectively.

Another way of thinking about the law of large numbers:

As above, repeatedly flipping n coins, a measure of how much the results spread from the expected number of heads, is proportional to the square root of n. This measure still grows as n grows. But if you want to express this as a percentage of the number of coins flipped, you need to divide by n, making it proportional to sqrt(n)/n. In percentage terms, the spread of the distribution goes to zero as n goes to infinity.

This has already been addressed a couple times, but the technical issue with your intuition is your sense of what's happening before you take the limit, and not with the limit itself.

You are first asking about a fixed number of flips N, and asking what is the probability of getting exactly N/2 heads (or tails). And then taking the limit of this as N goes to infinity. (Incidentally, "half the time" the probability is zero because you've flipped the coin an odd number of times and so it's impossible to get exactly N/2 heads (or tails). This observation may provide you with more intuition about what's happening.

But on the second one, you're asking about a fixed number of flips N, and asking what the mean of the number of heads (or tails) is over all possibilities (which is used to produce the 50% value). But this value is a constant, and so nothing changes when you're taking the limit.

If you saw 100 sequence H,T,H,T,H,T.... would you conclude an equally rigged coin is being used and bet on another heads?

When playing online poker, what would it take for you to suspect a rigged RNG?


PairTheBoard

Thx PairTheBoard, TimM, Aaron.




My original post should have stated "given a fair coin" ofc. Johns counter-question of H-T- repeating doesn't sound possible with a rigged coin in the way that all-heads might, but if the coin was computer-generated, say, would you suspect foul play after a long H-T- repeating sequence? (my guess is you'd say Yes).


An article I found while looking into all this described that one could usually distinguish between an imagined coin flip sequence that a subject writes out (they are told something like "imagine that you are flipping coins in your head and write down the results") vs. a real coin flip sequence that the subject actually performs and records. Typically, people imagine random sequences as being much more uniform than actual random sequences tend to be.
e.g.
Imagined sequence: H-T-H-H-T-H-T-T-H-T
Actual sequence: H-H-H-H-T-H-T-T-T-T

I had another idea about this that I like although it might not help people who don't know much probability. You can represent each of the i flips, i=1,2,... as random variables Xi which take on the values 1 for heads and 0 for tails. Then n flips are represented as a random n-tuple (X1,X2,...,Xn). An example outcome of the n flips might then look like (1,0,0,1,1,1,0,...,1,0,0).

The number of heads after n flips can be represented by the sum of the Xi's. i.e. S(n) = X1+X2+...+Xn. Notice S(n) is a random variable. It only takes on a specific value when the coin is actually flipped n times. Before you flip the coin n times you don't know what S(n) will be.

Just as you have a random number of heads in n flips, S(n), you also have a random average of the 0,1 Xi outcomes in n flips given by S(n)/n. Let's denote this random average by M(n) = S(n)/n . M(n) is called the "Sample Mean" for the Xi to distinguish it from the "Mean" of Xi. "Mean" is just another word for "Average".

We know the "Mean" for Xi is 1/2. We know this before we do the flips. The Mean is a fixed number depending only on the coin being fair and not on any specific outcome of n flips. But the Sample Mean M(n) is a random variable like S(n). We don't know what M(n) will be before we do the n flips. However, we can talk about probabilities for the Sample Mean M(n). For example, how likely is it that M(n) will come out to be close to the actual Mean, 1/2 ?

Now, what does it mean to say that the RANDOM Sample Mean, M(n) = S(n)/n "converges" to 1/2 as n gets large. Loosely speaking, it means that we know before the coins are flipped that M(n) will be close to 1/2 for large n with high probability. Or, a little more precisely, decide how close to 1/2 you want to get, for example somewhere within 1/2 +or- 0.0001. Also pick a probability as close to certainty that you want, say 99.999%. Then before we do the flips there is a large number N we can pick so that as long as we flip the coin more than N times the probability M(n) will be within 0.0001 of 1/2 will be greater than 99.999%. i.e. P(1/2 - 0.0001 < M(n) < 1/2 + 0.0001) > 99.999% for all n>N. And if we want M(n) closer to 1/2 with higher probability we just need to pick N larger.

Now here's the point to all this. We can handicap or advantage the number of heads S(n) by and number we want. We don't have to actually "see" 99 straight heads to start the sequence of flips. We can just specify an initial heads advantage to S(n) by adding 99 to it. i.e. look at the heads-advantaged random variable S(n)+99. Certainly, just choosing to add 99 to S(n) from the start should have no effect on S(n) or X1. And despite advantaging S(n) by artificially adding 99 heads to it from the start we will still see the same convergence for the advantaged S(n). i.e. the advantaged sample mean will still converge to 1/2.

(S(n) + 99)/n will still converge to 1/2 just like S(n)/n does.

In fact, you can advantage S(n) by any fixed amount and the same is true despite there being no effect on S(n) by the artificial advantage you give it.

The random advantaged mean (S(n)+1000000000)/n still converges to 1/2, not because S(n) decides to have more tails to balance out the artificial advantage it doesn't even know we gave it, but because the artificial advantage is drowned out by the law of large numbers for S(n)/n.

Large numbers can get very very large. Consider the googol. Then consider the googolplex.


PairTheBoard

Since a rigged RNG could be rigged to produce any outcome including an equal distribution of hands, I wouldn't be looking at aggregate, outcomes or their averages. Basically I'd look for predictive value in past events/results. For example, if a coin flipper was in cahoots with a coin flip contestant, the coin-flipping mechanic would communicate future outcomes through prior outcomes and that's what I'd be looking for.

I don't know precisely where I'd suspect something, though. But if I didn't know for certain that a coin was fair, if the first flip was heads, I'd say it's +ev to bet heads on the next flip on the chance the coin is unfair. It's basically a freeroll. Of course that edge is infinitely small at that stage but if the next toss is heads that infinitely small chance doubles and continues to double with each subsequent heads until the infinitely small becomes the infinitely large. So by the time we get to 99 straight heads it's virtually impossible the coin is not rigged or that tails is coming next.

Yea, that's about how I see it as well. When a drug company does a random drug vs placebo trial, the FDA wants to see confirmation of the company's predicted result for the drug, the "primary endpoint" of the trial. The FDA wants a small p-value like less than 0.005 i.e. half a percent. That's basically the probability the achieved result would occur by luck if the drug were no different than a placebo. The trial may also have secondary endpoints defined in it's protocol. In both kinds of endpoints they are evidence of the drug's efficacy the trial is on the lookout for ahead of time.

What is much less impressive to the FDA are patterns in the trial data that are noticed ad hoc, i.e. after the fact of the trial's run. Such ad hoc "results" need confirmation in a follow up trial where they may be looked at as primary or secondary endpoints. Some drugs are put through several such trials before the right conditions are found where the drug proves efficacy to the FDA's satisfaction.

The thing is, you can find ad hoc "results" in coin flips or commodity prices. It's called data mining. You can even buy software that does this kind of data mining for you based on historic prices of stocks or commodities. The software computes a trading rule which had you followed it over the historic period you would have made money based on the historic data. People then put the software rule in charge of their trading and hope to make money going forward. Other people make money selling the software.

For example, you might flip a coin 100 times and look at the results. You might notice that after HTH combos TTH came up more than it should have. And after HH combos, THH came up more than it should have. You might then bet on those successor combos at 7-1 odds (less house rake) whenever their predecessors came up going forward. But I would advise against it. Not only is it ad hoc but it's hard to think of any reason for it to have happened in the first 100 flips other than chance. It's not the kind of "rigging" I would be looking for ahead of time.

HTHTHT...HTHT through 100 flips looks pretty rigged to me though. My best guess is the coin flipper is a slight of hand artist who repeatedly switches 2-sided heads and 2-sided tails coins in and out of her flipping hand. Never bet against a magician.


PairTheBoard

UK magician Derren Brown's TV series had an episode about 'predicting' horse race winners, the episode title is The System (the full show is on YouTube, while it's 45 mins long I think it's worth the watch). It's not one of his 'hypnosis/mind reading' type of episodes, which I think are much less interesting, but more of the skepticism / scam-debunking, a la The Amazing Randie.

The lead in to the big reveal had him flipping a coin ten times in a row, with heads genuinely appearing every time. No camera tricks, no editing tricks, no trick coin. How he was able to do it was the clue to how his winning horse system 'worked'.

Good Show. The most impressive thing to me is how much work Magicians are willing to put in to prepare some of their "tricks". I don't see how he did the thing with the 4 people and the pictures. Doesn't seem like the same multiplicity method would work. Looks like some kind of forcing. I imagine the method was much slicker than what I could come up with.

I like the Penn and Teller "Fool Us" show. They don't fully reveal the methods but the hints they give give you a good chance of figuring it out. There was also a show on a few years ago that revealed the methods for many of the basic illusions magicians do. Probably a lot on Youtube.



PairTheBoard

This is a pretty common classroom activity in probability. You have students generate random sequences of coin flips (20 is enough, but something like 50 is much better), and then have a computer generate an equally long randomized sequence. You can guess the computer generated sequence by picking the sequence with the longest string of consecutive flips and be right a large percent of the time.

I also remember seeing a website that had a semi-machine learning thing where the user inputs a binary choice (heads/tails) that the computer tries to guess what you will pick. I think it just created a frequency table of all the k-tuples of choices you generated in the past and your choice immediately following that k-tuple, then guesses the most probable outcome. (And k is an embarrassingly small number, like 4 or 5.)

No coin is equally balanced. Or can be flipped in an unbiased manner forever. Of course through time (unlimited sequence) this converges.

Once my mom had to make up a list of dollar amounts representing random numbers of quarters. So she would write things like XX.25, XX.50, XX.75, but never XX.00 until I pointed out that 1/4 of her numbers should end that way.

But if a coin is unfair just because there's predictive value in the knowledge of prior results, other than looking for predictive value in prior results what options are left to render it more or less likely that the coin is unfair?

There's a difference between the coin being unfair and the experiment of flipping the coin being rigged. You can make statements about the probability the observed sample average would occur by chance with a fair coin. For example, seeing the first 99 flips all heads gives a sample average with extremely small probability of happening by chance with a fair coin. But what do you look for in a rigged experiment? I don't think you can suspect rigging based on any random outcome of flips. So what kind of outcome of flips would cause you to suspect rigging? That may be a deep question. Based on experience I guess. Sort of a, "I'll know it when I see it", kind of thing.

With the data mining software you get trading rules for various commodities based on historical data. You are then advised to test these rules going forward with imaginary rather than real money trades. You will find some of the rules fail to work going forward. You discard them. Then you begin trading with real money based on the remaining trading rules. As you go along you discard rules that loose you money and keep trading on the rules that make you money. When you run out of rules that make you money you go to work as a salesperson for the company that sold you the software.


PairTheBoard

James Randie used to put a small card in his wallet stating the day's date and that he predicted he would die that day, changing out the card with a fresh 'prediction' every day. Just so there'd be a mystery ("how could he know?!").

Yeah, I thought the same - I think it was probably put in as way to introduce all the Polaroids later in the show with the reveal, rather than being based on the same principle as the main story.

About the reveal:
[Spoiler] Most recent time I watched this episode, I realised the whole think was probably another level of a deception - meaning, it's much easier to pretend to have contacted a thousand people than to actually go through that process eliminating them bit by bit, hoping that the final person is suitable for a TV show. Plus, cell phone cameras were not widely available at that time, so how would they know who'd be able to make a video diary of the early events? No, it's much easier to pretend that all happened and have a second unit make the video diary part, plus a couple of extra 'failed' contestants.

It doesn't change the underlying story though, so not a big deal, if that's how they filmed. The woman was convincing enough that it hadn't crossed my mind at first, but if she'd really just bet what to her was a vast amount of money on a lie, she didn't react too badly!
[/spoiler]

If we plot the results of a fair coin against one biased to produce HTHT.... what we'll see will look like a movement to the mean chart with the biased coin representing the mean. I would not expect to see that. Nor would I expect to see the same lack of initial variance with a large trial broken down into smaller ones. But I need the sequencing to get there. If all I know after 1000 flips is 500 heads I have no reason to suspect the coin is unfair. On the other hand if all I know is HTHT.... after 1000 flips, I'm near certain the coin is unfair.
Yeah, I'm a long-ago retired play money trader. Just started looking like gambling for me once I started noting my reasons for a play and factoring out the times I was right for the wrong reason. But I've used similar tools and techniques with direct marketing. At the end of the day it's just about looking for statistical outliers and trying to prove via the negative that they're just that by ruling out every other conceivable and testable factor. Usually they're just outliers but sometimes outliers point to other variables involved or the emergence of a trend/movement of the mean.

You would never be able to prove rigging. However, what you would look for are deviations from the expected flips based on the sequence itself. And the amount of checking you can do is related to the length of the sequence. (See also your Bayesian prior.)

Suppose you have a sequence of length 1000. You can look at the H/T distribution to see if it deviates "too far" from the expectation. Then you can cut this into 500 pairs of flips and look at the HH/HT/TH/TT distribution, and check to see whether this deviates "too far" from the expectation. You can then do 333 groupings of 3, 250 groupings of 4, 200 groupings of 5, and so on. As you take longer subsequences, your data generates fewer of them, and so the strength of your conclusions gets weaker. Also, you need to correct for the number of inquiries that you're making. (See also "Green jelly beans linked to acne.")

So it's still possible to reach a quantitative conclusion.

The worst part of the data mining software for trading was that the rules it came up with based on historical data were all in a black box. You couldn't examine them to see if there was any plausible reason for why the data might have behaved that way and might be expected to continue in that pattern. It might not really be that helpful if you could do that (theories abound with traders) but it would at least provide a hint of credibility to the operation. When doing trials for a drug it's nice to have a theoretical mechanism of action.

That's not to say statistical analysis of data can't be useful. In fact, I guess that's the big new thing these days using up tons of processing power in the cloud. Big Data analysis.


PairTheBoard