Re: Probability, Evolution, and Atheism


"tonysin" <a2mgoog@xxxxxxxxx> wrote in message
news:ce97864d-b33c-4a41-afaf-2b46e8aec1fa@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Alvin Plantinga is a theology professor at Notre Dame, and he wrote
this essay on why belief in evolution is irrational:

http://www.christianitytoday.com/bc/2008/julaug/11.37.html

PZ Myers is a scientist who writes an excellent blog on science and
atheism called Pharyngula, and he wrote this entry to refute
Plantinga:

http://scienceblogs.com/pharyngula/2009/05/alvin_plantinga_gives_philosop.php#comments

In his essay, Plantinga uses an example of the extremely low
probability of flipping a coin 1000 times, and having heads come up at
least 75% of the time.

In his refutation, Myers scoffs at Plantinga's innumeracy, wondering
why he had to get help to do the calculation: "He needed help from an
expert to multiply simple probabilities? Does being a philosopher mean
you're incapable of tapping buttons on a calculator?"

Well, I agree with Myers that Plantinga's argument is lame, but I
disagree that the calculation is trivial. I would appreciate it if the
members of this group would help me out.

The problem is to find the probability that out of 1000 tosses of a
fair coin, 750 or more will be heads (I've changed the terminology a
bit, but not the math). My math is rusty, but it seems fairly clear
to me that the probability of getting *exactly* 750 heads out of 1000
tosses is the number of combinations of 1000 taken 750 at a time,
divided by 2 to the 1000th power, i.e.

(1000! / (750! * 250!)) / (2^1000)

I hope I'm right so far. Then the probability of getting exactly 751
heads is the same formula, but with 751 at a time, and so on. And so
the probability of getting AT LEAST 750 heads is the sum of the 250
terms for the probabilities of getting exactly 750, 751, ..., 1000
heads in 1000 tosses.

Am I right so far?

If so, I don't see a non-tedious way to add up the 250 terms, nor do I
know of a calculator that you can just "tap buttons" for numbers that
size. All of my calculators give an error for any factorial much
larger than 70!. Even my Microsoft Math 3.0 won't do combinatorics on
numbers that size. I imagine that a specialized program like Maple or
Mathematica would, but I don't have those.

So my question is, is there an elementary formula for summing those
250 individual terms to get the exact cumulative probability, or do
you have to use some approximation formula (like Stirling's formula
for large factorials) if you don't want to add them one by one?
Thanks for any help.

You are simply asking what the probability of getting n heads out of N flips
is.

This should be conceptually easy to understand. We just need to know how
many ways to come up with n heads is. That is, all the sequenes of possible
N flips that end up having n heads.

How can we do this? Well, what if we ordered the n heads from 1 to n then
the tails n+1 to N? This sequence is easy. Now to get all the other
possibilities are we not just randomly shuffling them?

e.g., suppose we are looking for 5 heads out of 7 total.

The sequence of HHHHHTT is very simple and easy to understand and is one
specific case. All other 5 heads out of 7 sequences will just be a
permutation of this, right? How many ways are there? permuting a set is as
that is simple. It is obviously N!. Here the only problem is that we do not
distinguish different heads so we must remove such indistinguishable cases.
Here it is n! for the heads and (N-n)! for the tails.

The result is there are N!/n!/(N-n)! total sequences that have n heads. Note
this is just the combinations of n heads in N flips.

Now the total number of possible sequences is 2^N.

Hence the probability of getting exactly n heads in N flips is
N!/n!/(N-n)!/2^N.

All the above assumes a fair coin of course. It's mainly for conceptulation
purposes and hopfully somewhat easy to follow. Again, the main point is to
realize that any possible valid result will just be a permutation of any
other valid result. Hence we can start with a simple valid result and figure
out how many ways there are to permute it.


Now to determine the probability for some n we just add up the
probabilities. It looks though as you got everything right so...


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