Flipping 10 heads in a row - a small probability demonstration | Hacker News Reader

Flipping 10 heads in a row - a small probability demonstration

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82 pointsColinWright40 comments

My favorite related demonstration is to do a "tournament" where you start with 64 people and eliminate the 1/2 who don't flip heads each round. Some "amazing" flipper will get through 5 to 7 rounds and win. Something to think about when you see a mutual fund manager who beats the market 7+ years in a row.

I have always liked to think of an international Rock-Paper-Scissors competition where everyone in the world participated.

One person in the world would win all of their matches in a row.

…and only play 40-60 rounds. Logistics aside it would be over in 5 minutes.

That's not quite comparable I think, with RPS being a game of skill and all.

There's a con based on this: send half your marks a prediction that the stock market will go up next week, and half a prediction it will go down. Repeat next week, restricting to whichever set received the correct prediction. After five weeks, you'll have a group of people who think you've accurately predicted stock market movements for five weeks in a row, and some of them will be prepared to pay for your next prediction.

I remember reading this anecdote to explain the anthropic principle:

http://beust.com/weblog/2010/08/17/fun-with-the-anthropic-pr...

That's really pretty neatly evil, I hadn't heard of that.

This reminds me of Derren Brown's The System: http://en.wikipedia.org/wiki/Derren_Brown#Derren_Brown:_The_...

Do you watch Leverage too?

(Leverage is a TV show in its 4th season, and last week's episode included this very con http://www.tv.com/leverage/the-boiler-room-job/episode/13919...)

It's not true that 1/2 the players will flip tails each round, or that there will necessarily be one winner. In fact, if you go 7 rounds with 64 people there's a 60% chance that no one will flip all-heads.

Yep, I should have said _about_ 1/2, but that's why I said 5 to 7 rounds. Even 5 straight wins produces the illusion of skill.

"Sir, you must be the greatest archer in the world - everywhere around on walls, trees and fences there are targets, each with your arrow exactly in the middle. How did you achieve such greatness?"

"I paint the targets after shooting."

A quicker way to do this would be to take a container full of 1024 coins and dump them onto the ground so they're spread out. Put all the coins that landed heads up back into the jar and continue this process 10 times. However many coins you have left are the number of coins that landed heads up 10 times in a row.

This is why probability is such a useful math: it turns our intuitive reasoning on its head and gives us solid descriptions of the processes at hand. Another great example of probability in action is the Monty Hall Problem (http://en.wikipedia.org/wiki/Monty_Hall_problem). Totally unintuitive process, but unarguable results.

I love the Monte Hall problem. I've found many smart people who have a difficult time getting it.

Another unintuitive probability (to me anyway) is the size of a group of people you need to have before it's more likely than not that 2 of them will share the same birthday

There's another very counter-intuitive one, but it requires a bit of a build-up:

Say there's a village that only has families who have 2 children each. Say the probability of any particular child of being a boy or a girl is 50/50.

First, there's a slightly-counter-intuitive question:

* Given a family where at least one of the two children is a boy, what are the chances that the other child is a girl?

Now, the super-counter-intuitive continuation, assuming you solve the above correctly.

* Given that no two children in the village have the same name, and one of the children in some family is named Joseph, what are the odds that the other child is a girl? (The answer is different!)

I'll bite.

For the 1st question, you are drawing only from families with boys (excludes families with 2 girls). Only a 1/3rd of picked families would have two boys, so 2/3rds have the girl as the other child.

For the 2nd question, if you are asking about whether Joseph's sibling is a girl, I think more information is needed.

How is the child named Joseph picked? If we assume that a random boy (regardless of family) is chosen to be named Joseph, 1/2 of boys have male siblings and 1/2 have female siblings (the boy-boy situation is double-counted if choosing a random boy), so it would be 1/2.

If we named Joseph by choosing a family with at least one boy and naming one of their boys "Joseph" randomly, you get the same 2/3rds as the 1st problem.

I don't see how knowing one of the children is named "joseph" gives us any more information than knowing one of them is a boy [1]. So I would be curious to see your logic for why the two have different answers.

[1] Unless you're leaving math-problem-world and want us to consider the probability that a child named 'joseph' is female.

Let's say we have a village with just 4 families, 1 for each possible case. The names of the boys are b0,b1,b2,b3 and the names of the girls are g0,g1,g2,g3 (as no two names are the same):

b0, b1

b2, g0

g1, b3

g2, g3

If we draw a random family that has at least one boy, we get all the families except the last one. And 2/3 of those have a girl in them, as opposed to 1/3rd that doesn't. So its 66% to find a girl.

However, if I ask you about a particular boy, I could ask about b0, b1, b2 or b3.

b0: sibling is b1

b1: sibling is b0

b2: sibling is g0

b3: sibling is g1

So for half of the possible names I choose, the sibling is a boy, and for half it is a girl.

Assuming the name was chosen at random from the existing names of the boys in the village, that makes the chance 50% to find a sibling girl again.

I just threw together a quick simulation of the Monty Hall problem if anyone's interested: https://github.com/Paton/Monty-Hall-Paradox-Simulation .. really fascinating.

I also have one for Matlab/Octave that allows you to change to door count to get a better feel for what's going on: https://gist.github.com/411593

So what is the world record of number of heads in a row? I fear this video could start a very time consuming coin flipping craze...

It's called Ion Saliu's paradox or The Fundamental Theory of Gambling. If you have an event with probability 1/n and repeat it n times, the probability of realizing the event at least once is is about 1-1/e = 0.63 (for high enough values of n).

His calculations were correct, showing that chance of failure was 37% and success 63%.

And also interesting to note that if you repeat n times, you will get on average 1 realization of the event (Sometimes 0, rarely n).

The Derren Brown episode he mentions is well worth a watch if you enjoyed this - he doesn't deal so much with the mathematical side, but the idea of using misdirection to show something unlikely happening simply through consistent trial.

Edit: the show is called "The System"

Ha! That's nothing. One time I got 12 yatzees in a row. One day I was playing Yatzee and got 2 in a row. That was exciting but I was impatient so I wrote a program to play for me. Eventually my program made 7 in a row but that was still taking too long. I calculated how long it would take to have a greater than 99% chance of getting twelve in a row and decided I needed to get a life more than I needed to keep the program running. Then I figured as long as it was really the computer doing it and not me it wasn't too much to imagine a Platonic universe where my program already had been run an infinitely long guaranteeing that I got 12 in a row. I don't have a video of it though so kudos to SingingBanana.

I've been listening to The Drunkard's Walk: How Randomness Rules Our Lives[1] by Leonard Mlodinow. It is a great introduction to how probability works, the history of randomness and probability in science, and how randomness affects so much of what we do. very interesting look at the math without getting too caught up in the math.

[1] http://www.amazon.com/gp/product/0307275175/ref=as_li_ss_tl?...

Actually, if you flip a coin 1024 times, the probability of getting 10 heads at some point is about 63%.

I was actually wondering about this myself the other day (https://plus.google.com/101522949595361604155/posts/Vhdist7x...), and so I build a little script to calculate it. Calculating the probability of a run is actually not very straight forward.

It's relatively easy toss a coin and always get heads (for both biased and unbiased coins): http://books.google.com/books?id=tTN4HuUNXjgC&pg=PA317#v... (Jaynes's Probability Theory: The Logic of Science, Section 10.3)

If Google is filtering those pages, here's an earlier version of that chapter: http://omega.albany.edu:8008/ETJ-PS/cc10i.ps

This is interesting psychology. Eventually 10 in a row has to happen. But would you bet 50 cents to win 51? At what point is it Bayesian statistics that tells you not to make the bet? (That somehow the coin is loaded)

You can assess the fairness or bias of the coin: http://en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair

If you wish to start with a probability distribution assuming a degree of fairness you can do that and it won't be so prone to fluctuation of the start. I don't know anything online about this offhand but check out "Data Analysis A Bayesian Tutorial" by D. S. Sivia section 2.1.1 for a worked out example.

Once you're doing that, then you need to calculate the optimum amount to bet. If the betting will continue indefinitely then a sensible thing to do would be to bet to maximise the expected logarithm of your bankroll, which you calculate with the Kelley criterion: http://en.wikipedia.org/wiki/Kelly_criterion

If you only get to make one bet then you may just want to maximise expected value.

EDIT: didn't explain how to translate the fairness distribution into win/loss odds because I don't know offhand, you could always simulate.

Amazing patience.

Two thoughts on this post/submission. The first is that it's really pretty shoddy, I mean why did it need to be filmed, why not a written (or even to camera) piece that just explains the logic. Anyone who doesn't believe it after having it explained isn't going to believe it after a video of an experiment that could so easily be edited to show 1000 heads out of 1000 flips.

The second is, even if it was presented better, is this suitable for HN? Sure, the topic of probability is suitable, but this is on such a basic level that surely most, if not all, HN readers already comprehended it.

That said, here's hoping that, if people are going to insist on upvoting it, it can at least spur some interesting discussion. With that aim, here's a tangeant:

It's always interesting in gambling how, no matter how mathematically smart someone is, it's incredibly easy to let your heart make decisions for you when money is on the line. Whether it's betting red on roulette because it landed black 6 times in a row, or betting big on a blackjack hand because you've lost your last few in a row and surely it can't keep going.

Even though, as you place the bet, you're thinking "I know the last X spins don't actually have any impact...", you can't help but feel the urge.

The video is definitely more engaging than text would have been. And I think it's a reasonable topic on HN. I knew some of it but certainly not all.

The video was visually similar to the set up that Derren Brown used in 'The System', which the video is making a direct reference to.

The person making it is trying to show how 'lucky' or 'unlucky' Brown was when it took him nine hours to achieve the same result.

I haven't seen the other video so didn't know it was done similarly. However my feeling is that it still isn't that relevant.

  Trying to show how 'lucky' or 'unlucky'...

He didn't really do that, did he? We know that it's possible to, on the first attempt, throw 10 heads in a row. Or it's also possible to not do that after 100 hours of trying. This video showed that it is possible to do it faster (be more 'lucky') than Derren Brown, but nothing about about the odds of when it might happen.

I just feel that you could take out 95% of the video, expand on the areas covered quite a bit, and have something much more interesting.

I like the philosophical implications of: if it is possible, with enough trials, it will happen.

I remember using this argument against the proposed creationism view of my religion teacher. If it is possible for life to emerge on a planet, given enough time, it is bound to happen. Likewise: given enough games of Go, one game will end with a perfectly ordered black-white spiral. This then led to the tangent of: Is it possible for God to exist in our world?

The same with the possible worlds model of modal logic. If something is possible, there is a world were such is the case. That could mean that, if time travel or travel between possible worlds is possible, there is a world out there, where they discovered this. They could travel between possible worlds and teach these other worlds how to travel too.

As soon as something that crosses the boundaries between worlds is even remotely possible, in the end it should permeate through all possible worlds.

> Is it possible for God to exist in our world?

This isn't the same kind of question, assuming the Abrahamic conception of God, because that God is logically inconsistent. So, if the Abrahamic deity exists, reality is inconsistent, so what right do we even have to consider it reality? Therefore, the Abrahamic deity cannot exist in the real world.

(Nope, no lightning bolts.)

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