The answer to this conundrum is almost childishly simple to our eyes. There are twenty seven ways of combining the possible sums to ten while there are only twenty five ways of doing so for nine: the trick is to realise that 3+3+4 is not the same as 3+4+3. Cardano was the first, by nearly a century, to figure out the mathematics behind this. Of course, he didn’t tell anyone – partly because mathematical tricks like this were regarded as trade secrets and partly, presumably, because he was using his knowledge to make money. This paper on the Complex and Unpredictable Cardano gives a fascinating review of his career and the issues discussed above.
Now, of course, the specific result of throwing any three dice is random but the statistical probability of throwing any particular triple is not. This insight, that surface randomness often hides deeper order is one of the most profound in the history of humanity and is really the whole basis of risk management.
Of course, with his mathematical prowess Cardano was in a position to honestly outwit his fellow gamblers – and also to detect when they were cheating. Yet he also realised something else of immense importance – that although he could calculate probabilities correctly he couldn’t adjust for the imperfections of the real world – so that, for instance, an imperfect die would yield imperfect results. Still, Cardano’s gambit – his willingness to take risks on his understanding of the perfect probabilities – seems to have paid off for him and he ascended to wealth and power.
Translated forward four hundred years many of the same insights apply to investors in stocks, although these are masked and obfuscated by the thicket of experts who spend their lives working on such things. The statistics of stock price movements are more random even than the throw of an imperfect set of dice and far more influenced by the malign manoeuvrings of card sharps with their marked cards.
Most investors today find themselves in the same position as Cardano’s gambling opponents did in the fifteenth century, groping around for some idea about how to manage and measure the randomness inherent in their chosen card game, yet with the overwhelming suspicion that there’s more going on than meets the eye. Playing Cardano’s gambit in this world is far, far more difficult than figuring out the probabilities behind a simple game of chance.
As all good gamblers know, there are occasions when the luck deserts you and there are times when you need to know when to fold and quit. Loss aversion, where people chase their losses because they can’t cope with their pain, is not something that affects such people. Playing the game means accepting losses on the grounds that the underlying probabilities will eventually fall in your favour.
Unfortunately probability seems to be something that’s inherently difficult for the human brain to get comprehend. This isn’t a minor issue confined to investors either. Consider the proposition that most CEO's are men. Well, even without hard evidence we’d likely accept that. Would that mean that we also agree that most men are CEO's? Hardly.
OK, now how about “most women with breast cancer have a positive mammogram”? Again, we’d likely agree. So what about “most women with a positive mammogram have breast cancer”? Not so, most women with a positive mammogram do not have breast cancer. Most diagnosticians don’t get this right when presented with the raw statistics – 1% probability that woman of 40 has breast cancer, 80% chance if she’s got it that the mammogram detects it and a 9.6% chance she’ll test positive if she hasn’t got it. The experts reckon that, based on these numbers, a woman with a positive mammogram has a 75% chance of having breast cancer.
They’re wrong, and not a bit wrong. They're massively wrong.
How many women with a postive mammogram will actually have breast cancer? You can find a discussion of this with the answers in the appendix to this paper, but the accuracy of diagnoticians improves dramatically.
This representational problem, technically known as base rate neglect, seems to be behind our fundamental difficulty in dealing with probabilities and is one of the factors that leads investors to make all sorts of strange gambles. In truth Cardano’s gambit is a simple one: understand the odds, minimise the risks and maximise the chances. In investing terms: analyse the fundamentals, manage you emotional biases and build the best margin of safety you can.
Cardano’s skills led him from poverty to riches and the support of some of the most powerful men in Renaissance Italy. Unfortunately he wasn’t so lucky with his children, who led him a merry dance before one of them betrayed him to the Inquisition. He ended his days back in poverty and certifiably crazy to boot. There is more to life than gambling, no matter how good at it you are.
The correct answer is about 8%: so at least 9 out of 10 positive mammograms are false positives.
You wrote: 10 out of 100 women at age forty who participate in routine screening have breast cancer.
Surely that should be 10 out of 1000? Otherwise, there is a 10% probability that a woman of 40 has breast cancer, not 1%.
It's not the stats that are the trouble, it's the proofreading ... Broken link to the reference paper also fixed.
A more nuanced view of share-price behavior, and of the matter of underlying order and surface randomness, comes from Benjamin Graham, Warren Buffett's mentor: "In the short term, the market is a voting machine. In the long term, it is a weighing machine."
According to wiki he secured lifetime annuity from Pope Gregory XIII. I don't think annuity form Pope count as poverty.
Cardano was tortured by the Inquisition in 1570, moved to Rome in 1572 and died in 1576. Gregory XIII received the pontificate in 1573 so the annuity was awarded in that period somewhere. That Cardano needed this at all given his former wealth and status is significant in itself.
Remarkably Cardano's autobiography is still in print, The Book of My Life. Extraordinary man, but I'm not sure he'd be someone you'd want to have as a friend.
Since I embarked on the journey of developing the Texas Holdem Investing theory I'm a big proponent of the similarities between investing and gambling (specifically poker in my case).
Great so see a luminary like yourself thinking along the same lines.