We'll have one more of these quizzes, but next time the species will be chosen randomly. At least then the success rate of Hugh Ross' proposals has a chance of reaching 50%.
1. Back in the day, I learned about the cool Monty Hall puzzle on a Usenet newsgroup. I found the puzzle to be very interesting, in that those who understand probability fairly well are most prone to being tricked by the puzzle. Here's the problem.
You're playing Let's Make a Deal with Monty Hall, and you are offered a choice among three doors. Behind one of the doors is a new Toyota Prius, yours to keep if you reveal it, and behind each of the other two doors is a goat (which, presumably, you don't want to take home). The game always proceeds as follows. You announce your choice of a door. Then Monty says, "Hmmm. Are you sure about that? Here, let me show you something that might change your mind." He then opens one of the two doors you did not choose, revealing a goat. Then he asks you: do you want to stay with your first choice, or would you like to change your mind? And the question is: do your odds of winning change (i.e., improve) if you change your mind, and choose the remaining door?
Now, if you've never encountered this famous puzzle, stop and think about it. I've put the rest of this section at the end of the post. Note, though, that there is no trickery here; Monty will always show you a goat (that's important) and the solution has nothing to do with semantics or other uninteresting chicanery. It's all about probability.
2. The evolution (and prevalence) of sex has long been considered one of the most perplexing phenomena in biology. Some of the most creatively-named hypotheses in all of science are hypotheses addressing the adaptive nature of sexual reproduction.
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| Image from PLOS Biology, photo by William F. Duffy. |
Enter the bdelloid rotifers, animals whose "scandalous" claim to fame is that they don't have sex. For centuries, it seems that the evidence that these microscopic animals are asexual amounted to the fact that no one had ever seen a male. A very nice recent review in Nature News explains how biologists have established that the bdelloids are actually asexual, and how these animals – alone among all others – pull it off. If you want more, PLOS Biology has an interesting review of how one famous theory of sexual evolution recently held up under duress.
3. Brain doping?! So, how many scientists are taking cognitive enhancers in order to outperform their competitors? And should federal granting agencies ban this practice, perhaps to motivate Major League Baseball to follow suit? (Beware of leftover April Fools jokes.) Some of my colleagues apparently do indulge in this practice. Some, I daresay, really should start.
4. I don't have much to say about Expelled, and I don't intend to spend any of my childrens' inheritance on it. (In fact, if you are contemplating such a foolish move, consider redirecting your expenditure in some more constructive direction.) But Chris Heard at Higgaion has posted a very important piece on Why Ken Miller isn't in Expelled. Check it out, and if you decide to waste synaptic activity on this issue, tune in to the NCSE's Expelled Exposed site.
5. According to Siris, philosophy is enduring a zombie invasion. David Chalmers must be pleased. I know I am.
6. Okay, back to Let's Make a Deal. The answer is: yes, you should change your guess to the other door. Your probability of winning is 2/3 if you do that.
When I first encountered the puzzle, I had a response that is typical among people who know a little about probability. I figured that Monty's little stunt with the goat is irrelevant; it couldn't change my chance of winning any more than any other silly behavior on his part. My chance of winning is 1/3, period. And of course I was partially correct. My chance of winning is indeed still 1/3 if I make my choice and just stick with it. But I was wrong in assuming that Monty's action is irrelevant. On the contrary, his goat-revealing gesture is determined by my choice. And it changes the situation entirely.
There are many ways to explain this, but here's my favorite. On average, 1/3 of the time I choose the car at the outset. In those situations, Monty gets to choose between the 2 goats, and I lose if I change my choice to the remaining goat. But 2/3 of the time I choose one of the goats at the outset. Monty is forced to reveal the sole remaining goat to me, and that means the car is behind the remaining door. So, I have a 2/3 chance of winning by randomly picking a door and then watching Monty show me the location of the car.
Why mention this on the blog? Well, for one, it's interesting. But also, this week the Monty Hall puzzle surfaced in a scientific context. Psychologists are debating the extent to which the Monty Hall phenomenon has affected the outcomes of numerous experiments examining so-called cognitive dissonance. Read about it at John Tierney's blog at the New York Times, and don't miss your chance to play the Monty Hall game yourself, especially if (like many others) you are unconvinced by my explanation.
3 comments:
I remember this MH puzzle! I went to the site and played the game 20 times, switching from my initial choice each of the 20 times. Percentage of victory? 55%, which means that I randomly chose correctly at the outset just under 50% of the time. Not exactly what I would have expected.
I was thinking about the Monty Hall problem this week because of this article: http://www.nytimes.com/2008/04/08/science/08tier.html?em&ex=1207972800&en=81bdecc33f60033e&ei=5087%0A My intuition on the problem was dead wrong. I was having a very difficult time figuring out why the two remaining choices weren't equivalent. The Wikipedia article did a nice job of pointing out the key wrinkle--as you did also. It points out to me that I overly trust my intuition.
The first time I heard the MH puzzle it fooled me as well. The best way I like to explain it is to increase the doors from 3 to 100, and then have the host open 98 door and show goats. Then it is obvious you want to switch because there was a 99% chance of being in that set of 99 doors, so to switch gives you overwhelming odds that are easier to see.
Actually, the first time I heard this puzzle it was with 100 doors, but the host only opened 1 door of the remaining 99, and I then had the choice to switch to any of the remaining 98 doors. You odds are still better, but your odds increase so insignificantly that it is much harder to notice. I asserted with confidence that the odds for any one door where still 1 in a hundred and I would not switch (and I was so proud of myself). I was wrong, if you switch, you improve your odds from 0.01 to (99/100)*(1/98) = 0.0101 (not much better odds)
bdelloid rotifers don't have sex? What would be the point to continue living!
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