Posts Tagged ‘The Monty Hall Problem’
Your correspondent is headed away for a week or so, ranging more then ten times zones from home– the current limit to continuous timely posting of (R)D… So, while regular service will resume on-or-around the 20th, a little something to keep one occupied:
Readers will recall that, on the occasion of an earlier hiatus, your correspondent wheeled out “the Monty Hall Problem” (c.f., “Riddle Me This” and “Birdbrains“). This time, with thanks to Prof. Stan Wagon at Macalester College:
Alice and Bob face three doors marked 1, 2, 3. Behind the doors are placed, randomly, a car, a key, and a goat. The couple wins the car if Bob finds the car and Alice finds the key.
First Bob (with Alice removed from the scene) will open a door; if the car is not behind it he can open a second door. If he fails to find the car, they lose. If he does find the car, then all doors are closed and Alice gets to open a door in the hope of finding the key and, if not, trying again with a second door.
Alice and Bob do not communicate except to make a plan beforehand. What is their best strategy?
Source: A. S. Landsberg (Physics, Claremont Colleges, California), Letters, Spring 2009 issue of The Mathematical Intelligencer.
As we craft our own strategies, we might solve a memorial problem for Gabrielle-Émilie Le Tonnelier de Breteuil, Marquise du Châtelet, the French mathematician and physicist who is probably better known as Voltaire’s mistress; she died on this date in 1749. Fascinated by the work of Newton and Leibniz, she dressed as a man to frequent the cafes where the scientific discussions of the time were held. Her major work was a translation of Newton’s Principia, for which Voltaire wrote the preface. The work was published a decade after her death, and was for many years the only translation of the Principia into French.
Judge me for my own merits, or lack of them, but do not look upon me as a mere appendage to this great general or that great scholar, this star that shines at the court of France or that famed author. I am in my own right a whole person, responsible to myself alone for all that I am, all that I say, all that I do. it may be that there are metaphysicians and philosophers whose learning is greater than mine, although I have not met them. Yet, they are but frail humans, too, and have their faults; so, when I add the sum total of my graces, I confess I am inferior to no one.
– Mme du Châtelet to Frederick the Great of Prussia
Readers will recall “The Monty Hall Problem”:
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
Over the years, the problem has ensnared countless people, including professional mathematicians. But not, it seems, pigeons. Walter Hebranson and Julia Schroder showed that, after some training, the humble pigeon can learn the best tactic for the Monty Hall Problem, switching from their initial choice almost every time. Amazingly, humans who get similar extensive practice never develop the optimal strategies that the pigeons pick up.
The original paper “Are birds smarter than mathematicians? Pigeons (Columba livia) perform optimally on a version of the Monty Hall Dilemma” from the Journal of Comparative Psychology is here… but the Discover article is a good– and for the humans among us, chastening– summary.
As we retreat to our fingers for counting, we might recall that it was on this date in 1833 that the first tax-supported public library was founded, in Peterborough, NH. The original collection consisted of about 100 books and was kept in Smith & Thompson’s General Store, which also housed the Post Office.