The Monty Hall Paradox
A ilttle while back on this blog I proposed that probability is fundamentally an epistemological matter. If someone had exact knowledge of the location of all of the particles in a "fair coin" as well as all of the relevant force vectors in a particular toss, then for that person the odds of heads coming up would not be 1/2 but 0 or 1. John G. in the comments section argued that while my contention is true, it is trivially so.
I think not, and as a paradigmatic case of why not, I offer the Monty Hall problem. In the TV show Let's Make a Deal, a contestant would choose among three closed doors. Then, often, Monty Hall (the host) would call for one of the unchosen doors to be opened, revealing the "booby prize" of, say, a goat. Then he would present the contestant with the option of switching her pick to the other, still-closed, door. Most people's intuition is that there is no statistical advantage to either sticking or switching -- the odds the contestant initially faced of any door concealing the top prize were 1/3, and that's still the case. But it turns out that that intuition is incorrect, and that the contestant interested in maximizing her odds of winning should always switch. Here's the explanation of why that is so from Wikipedia:
"The chance of initially choosing the car is one in three, which is the chance of winning the car by sticking with this choice. By contrast, the chance of initially choosing a door with a goat is two in three, and a player originally choosing a door with a goat wins by switching. In both cases the host must reveal a goat. In the 2/3 case where the player initially chooses a goat, the host must reveal the other goat making the only remaining door the one with the car.
"More formally, when the player is asked whether to switch there are three possible situations corresponding to the player's initial choice, each with probability 1/3:
"The player originally picked the door hiding goat number 1. The game host has shown the other goat.
"The player originally picked the door hiding goat number 2. The game host has shown the other goat.
"The player originally picked the door hiding the car. The game host has shown either of the two goats.
"If the player chooses to switch, the player wins the car in the first two cases. A player choosing to stay with the initial choice wins in only the third case. Since in two out of three equally likely cases switching wins, the probability of winning by switching is 2/3. In other words, players who switch will win the car on average two times out of three.
"The solution would be different if the host did not know what was behind each door..."
The alteration in the "objective" probabilities faced by the contestant hinges on the fact that Monty Hall knows what is behind each door, and is not picking randomly. It is the state of his knowledge that alters the contestant's odds.
Since many, many people have had a very difficult time accepting the correct solution to this problem, it suggests to me that the dependence of probability on states of knowledge is far from being an immediately obvious relationship.
I think not, and as a paradigmatic case of why not, I offer the Monty Hall problem. In the TV show Let's Make a Deal, a contestant would choose among three closed doors. Then, often, Monty Hall (the host) would call for one of the unchosen doors to be opened, revealing the "booby prize" of, say, a goat. Then he would present the contestant with the option of switching her pick to the other, still-closed, door. Most people's intuition is that there is no statistical advantage to either sticking or switching -- the odds the contestant initially faced of any door concealing the top prize were 1/3, and that's still the case. But it turns out that that intuition is incorrect, and that the contestant interested in maximizing her odds of winning should always switch. Here's the explanation of why that is so from Wikipedia:
"The chance of initially choosing the car is one in three, which is the chance of winning the car by sticking with this choice. By contrast, the chance of initially choosing a door with a goat is two in three, and a player originally choosing a door with a goat wins by switching. In both cases the host must reveal a goat. In the 2/3 case where the player initially chooses a goat, the host must reveal the other goat making the only remaining door the one with the car.
"More formally, when the player is asked whether to switch there are three possible situations corresponding to the player's initial choice, each with probability 1/3:
"The player originally picked the door hiding goat number 1. The game host has shown the other goat.
"The player originally picked the door hiding goat number 2. The game host has shown the other goat.
"The player originally picked the door hiding the car. The game host has shown either of the two goats.
"If the player chooses to switch, the player wins the car in the first two cases. A player choosing to stay with the initial choice wins in only the third case. Since in two out of three equally likely cases switching wins, the probability of winning by switching is 2/3. In other words, players who switch will win the car on average two times out of three.
"The solution would be different if the host did not know what was behind each door..."
The alteration in the "objective" probabilities faced by the contestant hinges on the fact that Monty Hall knows what is behind each door, and is not picking randomly. It is the state of his knowledge that alters the contestant's odds.
Since many, many people have had a very difficult time accepting the correct solution to this problem, it suggests to me that the dependence of probability on states of knowledge is far from being an immediately obvious relationship.
There should be a distinction between the nature of mind and thought generally and the nature of probability as a branch of knowledge. Any conclusions about the former will logically apply to the latter. If you conclude that the essence of Reality is mental, in some sense the "ground" of thought is essentially epistemological (to be known it must be knowable). This is not trivial in light of general inquiry.
ReplyDeleteHowever, if we are discussing the nature of probability qua probability, the ultimate nature of thought is somewhat beside the point. In economics and other sciences, probabilistic formalisms (or any formalism) often can be translated into one another if they are essentially similar, although some languages may be confusing or have very limited application. This is why context is important. The Monty Hall problem will be amenable to certain types of language while situations involving structural indeterminacy may require others.
So yes, probability is epistemological, but the important thing is to "do" epistemology, attempt to elucidate what we can meaningfully say with various probabilistic expressions and leave questions about the ultimate nature of knowledge aside.
There is an easy way to show that the correct solution to the Monty Hall problem is indeed correct. Just scale up the numbers ....
ReplyDeleteSuppose that instead of 3 booths, there are 1000. As with the original problem, only one has a car in it. The other 999 have goats. You choose one at random. The host then comes along, opens the door to 998 of the booths and shows them as all containing goats. This leaves your original choice and one other closed booth.
Should you switch?
Since this version of the problem has precisely the same logical structure as the original problem, the answer must be the same in both cases....
Julius
"So yes, probability is epistemological, but the important thing is to "do" epistemology, attempt to elucidate what we can meaningfully say with various probabilistic expressions and leave questions about the ultimate nature of knowledge aside."
ReplyDeleteJohn, I'm not sure why you contend this. What if I want to look at the ultimate nature of knowledge rather than doing epistemology. Is that wrong?
Julius -- your example is interesting, but it still only works if Monty's not choosing randomly. If he just happened to hit 998 straight goats, you'd be rationally indifferent about switching.
I've got my own version:
ReplyDeletehttp://urikalish.blogspot.com/2007/02/vertigo-and-lion-intuition-can-be.html
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