Chance News 21
Quotations
Forsooth
The next two Forsooths are from the October RRS NEWS.
Metro
16 March 2006
Times on line
20 March 2006
Estimating the diversity of dinosaurs
Proceedings of the National Academy of Sciences (http://www.pnas.org/cgi/content/abstract/0606028103v1), Published online before print September 5, 2006 Steve C. Wang, and Peter Dodson
Fossil hunters told: Dig deeper (http://www.philly.com/mld/inquirer/living/health/15439574.htm) Philadelphia Inquirer, September 5, 2006 Tom Avril
Steve Wang is a statistician at Swarthmore College and Peter Dodson is a Penn paleontologist at the University of Pennsylvania. This study was widely reported in the media.
In their paper the authors provided the following description of their results. Here are a few definitions that might be helpful: genera: a collective term used to incorporate like-species into one group,nonavian: not derived from birds, fossiliferous: containing a fossil, rock outcrop: the part of a rock formation that appears above the surface of the surrounding land
Despite current interest in estimating the diversity of fossil and extant groups, little effort has been devoted to estimating the diversity of dinosaurs. Here we estimate the diversity of nonavian dinosaurs at 1,850 genera, including those that remain to be discovered. With 527 genera currently described, at least 71% of dinosaur genera thus remain unknown. Although known diversity declined in the last stage of the Cretaceous, estimated diversity was steady, suggesting that dinosaurs as a whole were not in decline in the 10 million years before their ultimate extinction. We also show that known diversity is biased by the availability of .. Finally, by using a logistic model, we predict that 75% of discoverable genera will be known within 60-100 years and 90% within 100-140 years. Because of nonrandom factors affecting the process of fossil discovery (which preclude the possibility of computing realistic confidence bounds), our estimate of diversity is likely to be a lower bound.
To be continued
Probability theory is not all that useful
Don’t box yourself in when making decisions, John Kay, Financial Times, 22 August 2006.
In this article, John Kay, a weekly columnist for the Financial Times, outlines a variation on the Monte Hall problem, to highlight that human minds are not well adapted to dealing with issues of probability.
Suppose there are only two boxes and one contains twice as much money as the other. When you choose one, you are shown that it contains £100. Will you stick with your original choice, or switch to the other box?
Kay shows that it is easy to apply this problem to real situations:
Anyone who has changed jobs, bought a house or planned a merger has encountered a version of the two-box game; keep what you know, or go for an uncertain alternative.
In this game, players can lose only £50 but might gain £100 and they have no way of judging whether the £50 loss is more or less likely than the £100 gain.
Decision theory predicts an expected gain of £25 from an equal chance of winning £100 or losing £50. But many people dislike the prospect of losing £50 more than they like the prospect of gaining £100. Reflecting his economic background, Kay goes on to speculate that this irrationality may explain why the equity premium in finance is so high – volatile assets need to show much higher returns to compensate for the pain of frequently seeing small losses.
Kay then outlines what he calls the 'fallacy of large numbers'
If you accepted 100 gambles like this, you are virtually certain to end up with a substantial gain. But, you may say, I am not playing this game 100 times. I am only playing it once and you cannot guarantee a gain in a single trial. That is true, but it illustrates “the fallacy of large numbers”. On the 100th trial, you are in the same position as someone who is offered the chance to do it once. So you should not do it the 100th time. But then you should not do it the 99th time, or the 98th – or the first.
He concludes that probability theory works well for a limited class of problems, but the real world is much more open-ended and there is usually fundamental uncertainty about both the nature of the outcomes and the process that gives rise to them.
Questions
- Kay claims that 'the message of both the original Monty Hall problem and of this one is that, even in very simple cases, it is impossible to be certain that a particular mathematical representation of a real problem is a correct description. ... For people in business who rely on models and for people in financial services who must choose between boxes with uncertain contents every day, that is a disturbing conclusion.' Do you agree with his views? If yes, what does it imply for the teaching of probability?
- Repeating the game many times, increases the chance of achieving positive expected payoff. What hidden assumptions are being made here? (If a few more zeros were added to the payoffs, would your attitude be different?)
- Do you agree with his 'fallacy of large numbers'? What is it about the 99th attempt that makes different to the first attempt? (Perhaps, think of statistics instead of probability.)
- To benefit from the irrationality of the equity premium, Kay suggests to stop looking at share prices so often, so, in the long run, you will get the benefit of the higher return without the pain of observing volatility. Do you think that this will solve the problem?
Submitted by John Gavin.