That's a nice filter. (Of course, I'm a former mathematician as well.) Here's how I think of it:
- the prior odds that you picked the double-headed coin are 1/999.
- after seeing ten heads, the posterior odds that you picked the double-headed coin are (2^10)/999 - let's approximate this as 1. (Bayes' theorem usually gets expressed in terms of probabilities, but it's so much simpler in terms of odds.)
- so it's roughly equally likely that you have the double-headed coin or any non-double-headed coin; the probability of flipping an eleventh head is then approximate (1/2)(1) + (1/2)(1/2) = 3/4.
- the prior odds that you picked the double-headed coin are 1/999.
- after seeing ten heads, the posterior odds that you picked the double-headed coin are (2^10)/999 - let's approximate this as 1. (Bayes' theorem usually gets expressed in terms of probabilities, but it's so much simpler in terms of odds.)
- so it's roughly equally likely that you have the double-headed coin or any non-double-headed coin; the probability of flipping an eleventh head is then approximate (1/2)(1) + (1/2)(1/2) = 3/4.