Oh, and by the way - I was in the mood for some wholesome family entertainment, so I took a look at BAC's post:
To find the probability of a set of r specific values picked randomly from a distribution of n different values, we actually need to ratio the number of ways one can pick those r values from the distribution by the number of ways one can pick any r values from the distribution. Right?
For example, if we have a distribution with 5 possible values (call them a,b,c,d,e) and we want the probability of seeing c and d show up in a random draw of 2 values from that pool of 5 possibilities, we first need to find the number of ways we can draw c and d. Well that turns out to be r!, so the answer is 2 in that case.
Next, we need to divide by the number of ways one can draw ANY 2 values from the 5 possibilities. Note that drawing that value does not eliminate it from the pool. The formula to use here is nr. So there are 52 = 25 ways of drawing 2 values from a pool containing 5 different values.
So the probability of seeing c and d in a single observation in the above example is 2/25 = 0.08 = 8 percent.
So the formula I should have used in my calculation for the probability of seeing r specific values of z picked randomly from a distribution of n different values of z is
P = r!/nr.
As anyone with even basic mathematical competence (that bothers to read this trash carefully enough) can see immediately, this formula is totally wrong. r! grows much faster with r than n^r, so for large r, these "probabilities" become larger than 1. For a simple example, suppose there were only 2 possible values, heads and tails, and we wanted to know the odds of getting HHTT. According to BAC, that's 4!/2^4=1.5. Oops!*
Of course it's much worse than that - as I keep pointing out, BAC is using false
a posteori reasoning from the very beginning. His way of applying his wrong formula is a good example - he asks what the odds are of getting quasar redshifts close to some particular set of values, and finds that it's small. But he doesn't ask for the odds of finding some
other set of redshifts which are close to some
other set of those values - and yet, had that other set been the data, he would have claimed it to be equally unlikely and hence equally as significant.
*This formula is correct for
something - namely, the odds of drawing r
distinct values out of n - but that is not what BAC needs, nor is it how he uses it. There is no reason why two quasars can't have the same redshift, and in fact there is a case where they do in this same post.