A simple coin toss and all we know is that it is strongly biased.
What is the probability to get Head at the first throw?

Since we have no further information to favor Head or Tail, we have to assume
p(Head) = p(Tail) and since p(Head) + p(Tail) = 1 we
conclude that p(Head) = 1/2. Easy as pie.

But kind of wrong, because the only information we
actually do have about the coin is that p(Head) is certainly not 1/2,
because the coin is biased.


Anonymous said...

If I roll a fair 6 sided die, the expected value of that roll is 3.5, even though that is not a possible value for me to roll. Your statement strikes me as being very similar to that one.

wolfgang said...

I guess it is similar, but in your example one can easily see in which sense "the expectation value is 3.5" is actually true (sum of points divided by number of throws converges towards 3.5)

In my example it is not so easy to see in which sense "the probability is 1/2" can turn out to be true (for a particular coin).

RZ said...

so "knowing that coin is biased" is relevant starting on the second flip, while "knowing *how* the coin is biased" is informative already on the first flip.
I can live with that.

Cosma said...

To be fair to the Bayesian approach*, there is a difference between having a prior distribution which is a delta function at p=0.5 and a prior distribution which is symmetric (but not delta) on [0,1]. Both make the same prediction for the first flip, but they make different ones on the second and subsequent flips.

*: I can't believe I'm writing those words either.

wolfgang said...

RZ and Cosma,

>> on the second and subsequent flips.

notice that it is not known *how* the coin is biased.
Perhaps there is more than just a bias in the mean.
There could be a non-trivial auto-correlation.
It could be that the first toss is always H followed by a long series of T (or the other way around).

I dont know enough Bayesian statistics to know how the general case of a biased coin has to be handled, but I assume it would take a lot of coin tosses to figure it out...

Funny how this turns a biased coin somehow into a fair coin again...

wolfgang said...

By the way, with real coins there is indeed such non-trivial bias and "Stage magicians and gamblers, with practice, are able to greatly increase this bias, whilst still making throws which are visually indistinguishable from normal throws".

RZ said...

Sorry, I interpreted "A simple coin toss " as having only simple bias, no sophisticated auto-correlations.

wolfgang said...

It was my fault of course and
I changed the text.