# Consecutive coin flips, what is the appropriate statistical test for this word problem? [closed]

I was listening to a podcast by NDGT (Neil deGrasse Tyson, a prominent scientist) and he posed a simple thought experiment to illustrate the susceptibilities to cognitive bias. What I've come here to ask about is which rigorous statistical framework to use here...

# Question

What is the appropriate statistical test, which lead to NDGT's statistical conclusions, for the following thought experiment? and please explain. Thanks!

## NDGT's Transcript

Here's a good example. Line up 1,000 people. Give them a coin to flip. They flip the coin. About half will get heads. About half will get tails. If they get tails, tell them to sit down.

How many are left? 500. Flip a coin. Tails sit down. 250. Flip a coin. Tails sit down. 125. Flip a coin. Tails sit down. We're down to 60. Sit down. We're down to 30.

We're down to 15. We're down to eight, four, two, one. When you do this experiment, there's this one person at the end that flipped heads 10 consecutive times.

Now what happens? The press rushes to that person and says, how do you feel about this win? And here's a common response. I sort of felt that head's energy about halfway through. And I saw the heads of things this morning. And I knew I was going to win.

Oh, this is wonderful. And yeah, I knew. I felt today was going to be special. Very common response. No, not for this experiment, for people who win the lottery or something. Right. Did the press go to anyone else in that line and ask them how they felt? No.

So here's the bias. The bias is you think this person won, and that person thinks that person won, because of some sort of spiritual heads energy permeating their lives that day, when any time you do this experiment somebody is going to flip heads 10 times in a row. That's the nature of this experiment.

The person who flips heads thinks that they're special when it's just a statistical fact...

## Self-study attempt

I get that 2**10 = 1024, so like exponential decay, if you start with n(0)=1024 after 10 half-lives you get n(10)=1. So I understand the choice of 1000 people and 10 consecutive flips.

I was thinking it's a simple:

• $$n = 1024$$ trials
• $$k = 0$$ successes
• $$r = 10$$ runs
• $$p = \frac{1}{2^{r}} = \frac{1}{2^{10}} = 0.0009765625$$
• Probability that there are no successes →
• Binomial PMF: $$P(X=0) = \text{Binomial}(n, k, p) = 0.3676997394112712$$
• which implies: $$P(X\ne0) = 1 - \text{Binomial}(n, k, p) = 0.6323002605887288$$

"any time you do this experiment somebody is going to flip heads 10 times in a row... it's just a statistical fact"

Is this the best/most comprehensive test here for proving such a bold statement? This guy wrote a blog post on a multi-million trials simulation but got the opposite result (36.8% success, 63.2% failure).. I calculated P(X=1) = Binomial(n, 1, p) = 0.36805917219661977. These results seem suspiciously close to $$\frac{1}{e}$$.

Translating NDGT's statistical fact statement, is it failing to reject this null hypothesis?

• $$H_0 = k \ge 1$$

#### Question:

What is the most appropriate statistical test, which would lead you CV wizards to NDGT's statistical conclusions, for the above thought experiment?

What is the statistical test that would lead you to make the assertion "any time you do this... statistical fact"???

• The suspicious $e$ comes from this: $$\lim_{n\rightarrow\infty} \left(1 - \dfrac{1}{n}\right)^n = \dfrac{1}{e}.$$ This relates to the so-called martingale strategy; Wikipedia or this video will help you. May 16 at 8:00
• To reconcile your result with the result in Daniel Chadwick's blog, you've calculated $P(X=0)=0.368$, he's calculated $P(X=1)=0.368$, so that implies $P(X>1)=0.264$.
– Lynn
May 16 at 8:15
• A related and more general derivation for the limit $1/e$ to the one of @mjjjj is done with poisonnation. (it is actually the same principle, using $\lim_{n\rightarrow\infty} \left(1 - \dfrac{\lambda}{n}\right)^n = e^{-\lambda}$, but in a different form) May 16 at 9:15
• The question is a bit unclear to me. Your calculations are correct, so what is the problem? Is your question just about the 1/e limit, then just make the question only about that (and possibly such question already exists so first check that) and leave out the extremely large post with the story about NDGT. May 16 at 9:28
• NDGT's narrative is a way of describing statistical expectations -- what happens on average in many repetitions of the experiment.
– whuber
May 16 at 12:34

Heads to win, 1000 people flip coins, after 10 flips there is a winner every time

No, this is not true. It is not every time. As you computed the probability for one or more winners is $$100\% - 36.8\% = 63.2\%$$. The probability for exactly one winner is $$36.6 \%$$ as was computed in the blog post.

Whatever you use the statement 'every time' is wrong. (but that is besides the point)

But aside from that exact computation determining whether it is every time or not (which is besides the point), the point that is being made with this thought experiment is that something that happens with only a small probability is still likely to happen when you consider a large population.

(and sometimes it is even certain to happen, for instance in a lottery process that guarantees a winner by picking a lot number out of all entrees)

The thought experiment is used to point out how an event with a small probability is almost certainly gonna happen to at least one person given enough people that make a try. So the idea, that it happens to some person, does not make that person special as if they had special powers that caused the event to occur. But due to the bias, we sometimes consider unlikely random events that happened with some people as an indication that there was some special causal relationship.

Personally, I am not a big fan of this argument. Even as a statistician, or as a scientist, you can still be spiritual.

What Neil deGrasse Tyson is doing here, is mocking all those people that fall easily into the trap of the selection bias and believe that some winning event was due to some special personal situation rather than some random process.

However, it is not a good argument against the idea that the world is governed by some unknowable 'power' that is beyond the grasp of physics and statistical tests. Humans obviously have no magical powers like every time we say some magical spell something expected happens, like speaking Avada Kedavra is gonna incinerate some person every time again, but still, we can believe in a type of magic that is unpredictable.

If something happens to me like winning the lottery then I can 'believe' that is was due to something 'magical' and not just due to 'probability'. But, I may not know what it is that I did that caused that magical event to happen. Was it karma and should I be more nice to others? I don't know. In this respect we have to revert to Wittgenstein's final sentence in his tractatus logico-philosophicus

Wovon man nicht sprechen kann, darüber muss man schweigen.

• This brilliant answer was very helpful for me and was certainly answered in the style of my original question. Unfortunately my Q was closed for focus and I've since revised it. Thanks again for the help! May 17 at 6:24
• I do not see him mocking people. I see him critizing the press for only interviewing the winner ex post whilst disregarding all the counter examples. The press in the story is the place where a limited view at the facts leads to misleading storytelling. May 17 at 9:48
• @KardoPaska Even if the question is closed I think you can still "accept" an answer. May 17 at 9:49
• @Bernhard with the thought experiment he is making ridiculous the thoughts of the people that see things happen to them in their favor as if 'someone' is looking over them. He is using it in that context of 'believe in god' and he is also criticising (maybe mocking is too strongly expressed) the person that afterwards states 'kinda knew he was gonna win'. May 17 at 10:09
• In a world full of people claiming that you should get acupuncture for your pain, healing crystals for your karma and turn to the true religion, each substantiated with case reports, we need voices of reason to explain that an apparently miraculous event does not need a miracle to happen. I fail to see the ridicule in that but that may be me, reading this in a foreign language. May 17 at 10:43