Did global warming help make Harvey and Irma as dangerous as they became? Hard to believe we have an administration that won't even consider the question and keeps busy eliminating "climate change" from research papers. Here's a lengthy list cataloging Trump's war on science.
Tesla temporarily upgraded to its Florida Owners' cars giving them an extra 30 miles of battery life. Glad they did this but it begs the question why Tesla restricted the battery life in the first place. Reminds of when in the 1970's you wanted a faster IBM computer, you paid more and an IBM technician would come and turn the appropriate screw. Competition prevents software-inhibitors to hardware. Who will be Tesla's competitors?
During all this turmoil the follow question by Elchanan Mossel had me oddly obsessed: Suppose you flip a six-sided die. What is the expected number of dice throws needed until you get a six given that all the throws ended up being even numbers? My intuition was wrong though when Tim Gowers falls into the same trap I don't feel so bad. I wrote a short Python program to convince me, and the program itself suggested a proof.
Updates on Thursday: I never did lose power though many other Georgia Tech faculty did. The New York Times also covered the Tesla update.
@lance. I liked your explanation the most. But doesn't that mean that we get the same answer for .... the question .... what is the expected number of dice throws needed until you get a six given that all the throws ended up being *odd* numbers?
ReplyDeleteBy symmetry I suppose ?
If all throws are odd you'll never get a six. If you want odd throws before the six, then there are only three possibilities before the 6 (2,4,6) that would make the sequence have length one so I'm guessing the expected length would be 2.
DeleteThe question by Elchanan Mossel also fascinated me. My intuition was totally unable to understand the correct answers or their explanations, before I read Ido's suggestion: "Suppose you throw a die with with 1,000 sides instead of 6". Which is funny, because I used this same intuition boost, when trying to convince people of the correct solution to the Monty Hall problem.
ReplyDeleteFor me, this hints that probability theory could have been a good example of Werner Heisenberg's analysis of the gap in Immanuel Kant's a priory of the categories of thought. And contrary to relativity theory and quantum theory, this example was already available at Kant's time. The abilities of our mind to grasp probability theory are underwhelming, and yet it is an important category of thought for understanding the world around us.
(Don't underestimate Werner Heisenberg as a philosopher! His account of closed theories already contained central ideas of Thomas Kuhn's later elaborations on paradigm shifts. But my description above is underwhelming, since somehow space and time don't occur in Kant's categories of thought, but Heisenberg's analysis actually focused on those.)
The fact the probability theory lends itself for exposing surprising examples of failed intuition has bothered me since high school. The following is an excerpt of an email I wrote to Rudolf Taschner (in German) after reading his book 'Game Changers: Stories of the Revolutionary Minds behind Game Theory' (or rather his German book 'Die Mathematik des Daseins: Eine kurze Geschichte der Spieltheorie'):
ReplyDelete"""
Diese Rezensionen beschweren sich nämlich, dass Ihr Buch zu viel über Wahrscheinlichkeitstheorie, Philosophie und Personen, und zu wenig über Spieltheorie geht. Aber gerade die Verbindung zwischen Wahrscheinlichkeitstheorie und Spieltheorie war für mich persönlich sehr erhellend. Die Wahrscheinlichkeitstheorie war also ursprünglich eine Spieltheorie, und die Spieltheorie kann nur Mithilfe der Wahrscheinlichkeitstheorie die Wirklichkeit sinnvoll modellieren.
Nachdem wir in der Schule Wahrscheinlichkeitsrechnung gelernt hatten (irgendwann in den 90er Jahren in Baden Württemberg), rechneten viele meiner Klassenkameraden mit "felsenfester Überzeugung der Richtigkeit" die falschesten Wahrscheinlichkeiten aus (nicht nur im Unterricht). Ich konnte mir kaum vorstellen, dass sie dies vorher auch schon getan hätten. Vielleicht hätten sie deutlich länger überlegt und viel länger gebraucht, aber sie hätten nicht so falsche Ergebnisse produziert, und nicht so felsenfest an diese falschen Ergebnisse geglaubt. Vielleicht wäre es viel sinnvoller, in der Schule gewisse Teile der Spieltheorie zeitgleich mit der Wahrscheinlichkeitstheorie zu unterrichten. Dann wären die Wahrscheinlichkeiten weniger abstrakt, würden sich sinnvoller interpretieren lassen, und die Schüler würden sich weniger dogmatisch an irgendwelche sinnfreien Regeln klammern.
"""
Especially in the last paragraph, I talk about "adamant conviction in those wrong results" and "dogmatic belief in senseless rules", which shows how this observable phenomenon of failed intuitions (about probability) of normal humans fascinated me.
Re. Tesla, my guess is there is a knob that trades-off torque vs efficiency. There are analagous "knobs" for combustion engines that, e.g., VW was manipulating. I guess you can argue that Tesla should have been more transparent and said "we are giving 30 extra miles and adding 1 second to your 0-60 time"
ReplyDeleteYea I got 3/2.
ReplyDeleteAlso I hope all is well there. An interesting commentary I heard recently on catastrophic climate change with respect to the hurricanes is that the opposite question is more relevant: how can we not attribute monster hurricanes to the changing climate? Warmer air holds more moisture than cooler air, a simple fact requiring genius to overlook.
ReplyDeleteWhat will be the number when the number of sides of die is 8 - rest of the conditions being the same (all the outcome is even and you stop when you get 6)
ReplyDelete