The sky is cloudy again, and it’s snowing again, too. “The snow was beautiful this morning!” exclaimed Denise Holland before quickly admitting, “but I wish it would go away!” Today’s picture is from yesterday’s weather, a shot south-southeast from Omena across the icy bay, Arctic in appearance but bathed in sunshine. Bob Bergdoll from Omena (who most recently finished and highly recommends PEOPLE OF THE BOOK, by Geraldine Brooks) stopped in to distract me from the puzzle on Monday afternoon with his clear, simple, stunningly elegant proof of the probability problem. In essence, Bob’s chart is a truth table, of the sort beginning formal logic students learn to construct, with WIN and LOSE in place of TRUE and FALSE. There are three doors, therefore nine possible combinations. Where is the car, and what door is selected? Bob’s introduces columns are for “Stick with Selection” and “Switch Selection,” and all you have to do is add up the WINs in each column to see the result: three in one, six in the other. He made up the chart in preparation for writing a computer program, but the chart itself tells the story, making the program redundant.
Oh, the joys of logic! Teaching my first class in the subject at the University of Illinois when only a nervous graduate assistant, I first realized the power and comfort of this dry field. Three smart young men sat together by the windows, waiting for me to stumble. When one of them doubted that affirming the consequent was a fallacy (“Are you sure?” he pressed, eyes narrowed challengingly), what wonderful calm settled over me as I said to him, nodding, “Do the truth table.”
8 comments:
AAARGH! I'm mostly sick of the probability of more snow. However . . .
When a chooser looks at three doors and knows that there is a car behind one of those three doors - not over there under the tree, or back behind the house, or anywhere else but behind one of those three doors - the chooser has a one-in-three chance to open a door and find the car. Once the door is opened, the chooser has found a car or not, but that situation is over, done with, dead.
Assuming there was no car behind the first door selected, the chooser is confronted with two doors, and the assurance that a car lies behind one of those two doors - not over there under the tree, and not behind that door that's hanging open, but behind one of these two doors. At that point the chooser has a ONE-IN-TWO chance of opening a door and finding a car.
There is, of course, another possibility. There has never been a car, and the chooser is doomed to disappointment. This is called spin and is fairly common in certain circles.
I know, I know! I argued like this for days, as did many much logicians and mathematicians much smarter than I am. In the end, however, these big brains said, "Oh, yeah, right." The table makes it clear to me, though my "common sense" still revolts.
On the question of probability of more snow, you and I have no argument!
I posted this earlier, but it seems to have vanished, so I'll try again. Sorry for the length.
(And yes, I'm tired of snow, too. But I didn't really want the 2 to 3 inches of rain we got here in Woodstock today, either.)
I'm afraid that Gerry has given us a different game, which of course has a different answer. In Gerry's version, it seems the contestant gets to pick a door, and open it. That had a 1 in 3 chance. Then, the contestant picks one of the remaining two doors, and that choice does in fact have a 1 in 2 chance.
That was really two games. Game 1: Pick 1 door from 3 and open it. Then game 2: Pick 1 door from 2 and open it.
But the original game did not have the contestant open the first door. It was a single game of two actions by the contestant, with an action by the host in between. First the contestant chooses a door, then the host opens a different door, then the contestant opens either her original door or a different one. And in that case, sticking with the original door is a 1 in 3 chance, while switching is a 2 in 3 chance.
Perhaps getting away from doors will help. Consider 3 sealed envelopes (red, green, and blue) with one of them containing a slip of paper that says "you won".
The contestant chooses one envelope, and marks it. The host then puts that envelope into a goldfish bowl, and puts the other two envelopes into another goldfish bowl (it helps if the bowls start out empty :-)
At this point, the probability is 1/3 that the bowl with the contestant's envelope has the prize in it (call this bowl 1), and 2/3 that the other bowl, with two envelopes (bowl 2), has the prize.
The host, knowing which envelope has the prize, now opens an envelope in bowl 2 that does not have the prize. He shows us it does not have the prize, and puts it back into bowl 2.
Before he did that, we knew that with probability 2/3, one of the envelopes in bowl 2 had the prize. Even though we now know that one of the envelopes in bowl 2 does not have the prize, that knowledge does not change the probabilities. The probabilities were set when the host put the envelopes into the bowls.
So, the probability is still 2/3 that one of the envelopes in bowl 2 has the prize. And since there is only 1 envelope there whose contents we do not know, the probability must be 2/3 that that envelope has the prize.
Absolutely right, Walt. Gerry, you didn’t get to open the first door you picked. But Walt, I’m afraid I like the idea of putting the envelopes into a goldfish bowl FILLED WITH WATER—AND FISH! Maybe I’m just getting punchy after thinking about this problem for so long, but when I read “it helps if the bowls start out empty,” I couldn’t help laughing. “What’s so funny?” David called out. “Oh, just a comment Walt wrote on the blog.” When I told him it was more about the doors, though, he lost interest. Rain? You had rain? The closest we got to that warm was icicles dripping in the sun.
I'm gonna grab a goldfish bowl and start beating a dead horse with it.
Which would you rather have, that car the host is trying to keep you from finding or a stinkin' slip of soggy paper that says "You won" in streaky ink, eh?
But Gerry, the host likes people to win (at least sometimes) because that keeps other people watching and dreaming! Which would you rather have today, more sunshine and COLD or warmth and RAIN? We'll get what we'll get, as always, but your welcome is always warm at Dog Ears Books--not to be too crassly commercial.
Well, I tried this out and discovered it really didn't make much difference which goldfish bowl selection I make: if you throw lake-effect snow into the equation, because that adds a fourth dimension?
I was even more impressed with your photos. I looks just like it does out our back door. This is the first time I can remember Lake Michigan freezing down here (it comes and goes). We are inundated with snow which makes me feel that I'm in Nport. However, we pray for more snow down here, rather than rain, because the ground is so frozen the rain has nowhere to go except all our basements! Need a good thaw first: an new eaves troughs, I guess.
It was -6 degrees this morning in Traverse City, where I had to go to teach. Our car almost didn't start. Everyone in Northport, even the most Pollyanna-ish, is tired of the cold!
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