
The ‘Hot Hand’ Debate Gets Flipped on Its Head - ssivark
http://www.wsj.com/articles/the-hot-hand-debate-gets-flipped-on-its-head-1443465711
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DennisP
Here's the paper's first paragraph, matching what the article said:

"Jack takes a coin from his pocket and decides that he will flip it 4 times in
a row, writing down the outcome of each flip on a scrap of paper. After he is
done flipping, he will look at the flips that immediately followed an outcome
of heads, and compute the relative frequency of heads on those flips. Because
the coin is fair, Jack of course expects this empirical probability of heads
to be equal to the true probability of flipping a heads: 0.5. Shockingly, Jack
is wrong."

But actually Jack is right. Here are all the possibilities. A "streak" means a
head was followed by a head, and a "break" means it was followed by a tail.

    
    
              Streaks Breaks
      TTHT    0       1
      TTHH    1       0
      THTT    0       1
      THTH    0       1
      THHT    1       1
      THHH    2       0
      HTTT    0       1
      HTTH    0       1
      HTHT    0       2
      HTHH    1       1
      HHTT    1       1
      HHTH    1       1
      HHHT    2       1
      HHHH    3       0
      total   12      12
    

The paper's argument later is more complex than the first paragraph implies,
putting the sequences in groups with fixed numbers of heads. I don't see the
point of that but it doesn't seem to help.

They get into some math, but just counting the cases doesn't seem to support
their argument at all. If anyone can explain their argument in a simple way,
I'm interested.

If they're really confident in this, they should go to Vegas, find a high-
rolling gambler, and start betting on coin flips. After each heads, offer
55/45 odds that the next coin flip will be heads. I'm sure it won't be hard to
find takers.

~~~
Beltiras
See my MC in a post below. I think I understand what they are on about. It's
biased against repeating in short runs.

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ssivark
The actual paper:
[http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2627354](http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2627354)

~~~
clarkevans
This seems similar to the Monty Hall problem, which is also quite unintuitive.
Perhaps what makes it work is that the person doing the predicting sees the
sequence as a whole, and hence, future events are dependent upon historical
ones -- and are no longer purely random.

Consider this question, "You're _somewhere_ in the middle of a 4 coin toss,
and the last toss came up heads, what's the probability of the next one coming
up heads?" I think the paper is saying it's a 40% chance -- you know you're in
an finite series, and, you have partial information about that series.

~~~
DennisP
That's what I counted, and got a 50% chance.

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IkmoIkmo
Is this a joke or am I the only one who thinks this is a ridiculous article?

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Beltiras
Very naive Monte Carlo:
[https://gist.github.com/arnists/228c4e77b1e2aa6d33f1](https://gist.github.com/arnists/228c4e77b1e2aa6d33f1)

Consistently comes back around p. 50 runs have a span of .25%.

Haven't read the paper yet, but if the PRNG isn't broken, I'd say it
invalidates the naive presentation at the start of the article.

EDIT: I think I understand the fallacy the authors present. This holds true
for short runs. E(H|H) _will_ be lower in short runs but asymptotically
approaches p when number of trials rise.

~~~
gtr
I enumerated the sequences that the article mentioned, and counted how often a
tail filled a head, and vice versa, and got 12 instances where a head follows
a head, and 10 where a tail followed a head. So there is a difference for just
counting up all the possible 4 flip sequences where at least one of the first
three is a head. However, doing a randomised test where I generated a random 4
length sequence, rejecting it if none of the first three was a head, then
doing the same test showed no real difference.

Code here
[https://github.com/gregryork/Flips/tree/master/src/flips](https://github.com/gregryork/Flips/tree/master/src/flips)

~~~
Beltiras
Take a look at the updated gist. There's an asymptote for P(H|H) when trials
grow approaching p.

EDIT: Graph is empirical H|H against run length (output of the last function
in a linear graph)

~~~
gtr
I'm not really sure what I am looking at there. Is the X axis the number of
trials run, or the length of the run?

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darkxanthos
I think that this is flawed in the sense that most people (including the
article's author) might misinterpret it. The paper assumes that we know when a
streak ends. So given that we have had a streak of tails and that it's broken
what's the probability of heads? That will be biased toward heads. I've only
skimmed the paper, but maybe they ultimately mean this is the source of the
incorrect bias in the layman's intuition?

