The example about the taxicab colour seems OK.

the calculationa about PSA and prostate cancer are hopelessly confused.
(1) Both 88/90 and 0.7 are probabilities of NOT having prostate cancer, under certain conditions, so the 88/90 doesn't belong as a factor in a calculation for hving prostate cancer.
(2) Both 0.927 and .07 are also both probabilities of not having prostate cancer, under certain conditions - so you shouldn't be multiplying them together!

readers should be aware the OLO has NO editorial standards WHATEVER - erroneous factual material is considered justified on the grounds that it's a site for writers "opinions"

Chris, A high PSA score, on its own may not mean very much!
Digital rectal examinations, may be more reliable, however uncomfortable or unreliable?
Some very nasty cancers can still be missed, given only so much of the prostate can actually be examined by this method. MIR scans, miss very little!
Even were you to have a prostate cancer, the very latest treatment, where no actual surgical intervention is required, just very well aimed very narrow beams of radiation, assisted by some chemo, reportedly seems to have better than average results, without the usual side effects!
If you have any concerns whatsoever, you should see your GP ASAP, and discuss your concerns with him.
If you are not completely certain his knowledge is contemporary, ask for a second opinion, a third; and or, a specialist referral. Castration as preferred by some, so called specialists, seems to have no better outcome, based on statistical evidence, in terms of life expectancy, than simply living with the problem?
See your Doctor!
Cheers, Rhrosty.

Chris,
It was good to hear that I am not the only one that is going through this ordeal.
My PSA shot up and a biopsy showed a Gleason of 3 + 3 = 6
A CT scan then showed that I had a lump on my kidney.
I am now at home recovering from an open left partial nephrectomy.
I see my urologist on Monday about my radical prosectomy.
Your Bayes calculations are interesting but one statistic is certain.
You had prostate cancer with 100% certainty.
Thanks to medical science you can now live a happy life.(even though probably sexless).
Five years ago this would not have been diagnosed till it reached your bowel. (From which my father died and the reason for the PSA tests.)

@ Jeremy
I am glad the taxi example was ok - it was taken verbatim from Silver's book.

Would you kindly post the calculations as you think they should be done.
I will revise the article and give you the credit.

I would not criticise Graham/OLO for just posting opinions. It does not purport to be a scientific journal with peer review boards, etc. What it relies on is readers like you to come up with corrections if needed. OLO does not have the resources to have a fact checking unit.

What happened in this case is that I posted the initial version of this blog in blogspot and circulated it to the 600 or so contacts that I send a monthly joke compilation which is totally politically incorrect.

Subsequently 5 of the contacts sent me an email saying my article had either persuaded them or their partner to have a PSA test. On the basis that if one person's life is enhanced it is better than none Graham agreed to post the article.

Again I thought I had got the calculation correct but as I said this was the first time I really sat down and tried to understand Bayes Theorem and I admit as a first time user my comprehension may be inadequate.

Again please post the calculations and I will update the article.

EQ,

You seem to be writing as though you are the author of this article - is that correct?

I don't "criticise Graham/OLO for just posting opinions", I criticise OLO for posting purportedly factual material without any interest whatever in whether it is accurate or not.

Who is Graham, anyway? Is he involved in running the site? He wouldn't by any chance be a person using the pseudonym "GrahamY" who has previously defended the site's editorial policy? As I recall, that poster defended publishing false material on the ground that other (allegedly) false material had appeared on the same topic.

And if the site is no better than a blog on which anyone can publish anything, then it should be promoted as such. Inexplicably, it gets some worthwhile authors (eg Andrew Leigh).

To return to your request that I should do the your job in regard to Bayes' theorem, in the taxicab example you want the relative probabilities of (1) silver, identified as silver, and (2) black, identified as silver. Use Bayes' theorem.

In the PSA example, you want the relative probabilities of (1) positive test, no PC and (2) positive test, PC. The information you provided says directly that this is 3/10. Bayes' theorem is a red herring

Obviously this is a matter for personal judgement, however
Dr Richard Ablin is the man who discovered PSA, and he says that PSA testing has become a "hugely expensive public health disaster," I think that Simon Chapman has also expressed a view that PSA screening is a waste of public money and results in a lot of unnecessary treatments and inevitable medical induced harm.