Good analysis. But what happens when the distribution of envelope amounts is unbounded? Note that with your geometric distribution, the expected gain for switching is still 25%, UNLESS you're at one endpoint of the distribution or the other. So if you spread out the distribution to make the endpoints infinitely unlikely to be reached, does that change anything?
I should have stated the problem more carefully; the amount in the smaller envelope is also a random variable. (E.g., you don't know if the envelopes contain $0.01 and $0.02, or $1 and $2, or $437 and $874, or $1 trillion and $2 trillion.) All you know is that you're either doubling your money by switching, or halving it, with equal probability. So why is the 125% expected return calculation (1/2 (0.5) + 1/2 (2)) not valid? Clearly if you knew in advance that the two envelopes contained exactly $5 and $10, your argument would work. (Because the larger the amount in your envelope, the less chance the other envelope has more.) But when the amounts in both envelopes are infinitely variable, except for the known 2:1 ratio, the answer is a bit more subtle. For example, suppose you peek in your original envelope and it contains $100. You don't have any prior information about whether the other envelope contains $50 or $200. So why wouldn't you switch? Does peeking in the first envelope actually give you any useful information? The puzzle is not as obvious as it seems.
Yes, but you're missing the REAL puzzle, which is that even the FIRST switch (with the calculated 1.25x expected return) doesn't gain any information. By symmetry, the expected return on the initial switch MUST be exactly 1.0x, yet the simple math says 1.25x. Where does the math/logic go wrong?
Suppose Monty Hall gives you a choice of two envelopes. Each envelope contains a check, and one of them is written for TWICE the amount of the other. So you pick an envelope.
Now, Monty gives you the chance to switch envelopes. (Assume Monty always gives you the chance to switch.) Logically, since your envelope contains X, the other envelope can contain either 0.5X or 2X, with 50% probability... So the expected value of switching envelopes is 50% (0.5X + 2X), or 1.25X. So, you should switch.
But here's the tricky part: Monty now gives you the chance to switch back! Since your new envelope contains Y, then by the same logic as above, the expected value of switching back is 1.25Y... So you should switch back. Right?
Clearly, something is wrong with this chain of thinking. Can you figure out what it is?
In the latter scenario, it may be our goal simply to preserve our initial odds. If so, it pays to toss a coin on the second choice. This way, quite regardless of the host's strategy, we will have our odds at 1/3 or above.
No, your odds with flipping the coin to choose between two doors will always be exactly 1/2, regardless of the host's strategy. (Think about it.) But if you know something about the host's strategy, then a non-random choice can give you better odds; 2/3 from switching in the classic version.
However the 6-bit screen only puts out 63+63+63+1 = 190 different colors.
It's technically still 64 + 64 + 64 = 192 different colors. That's because the contrast ratio is finite; the "black" setting yields dark red, dark green, and dark blue colors on the subpixels, respectively.
I wrote an article on my blog with some related thoughts about solar. In particular, I've considered installing solar panels on my roof, but my geographical region has a tendency to accumulate dust very quickly, so I'd be out on the roof cleaning the panels all the time if I were to have any chance of breaking even.
Or, you could buy yourself a big air compressor for about $400 at Lowe's, and set up some compressed-air lines and nozzles over your solar panels. Connect this to some sort of electrically-powered solenoid valve and a timer, and the compressed air will blow off the dust for you. Of course, this will take a small amount of power, but it's a lot easier than climbing onto your roof every week or so.
The thing is, with home-installed solar, the margins are so narrow that it doesn't take much for it not to be profitable anymore. And when you factor in condensation at night, the dust gets rather caked on; like a car windshield left outside. You can't just blow the dust off; it takes at least soap and a squeegee. In any case, this doesn't help the high cost of actual installation of the panels in a custom location, and wiring it into the house electrical grid, plus the time and expense dealing with permits, etc. For new construction, solar might make sense; but for retrofitting, it's doubtful. However, I think the co-op type idea could really work, and hope somebody makes it happen.
I wrote an article on my blog with some related thoughts about solar. In particular, I've considered installing solar panels on my roof, but my geographical region has a tendency to accumulate dust very quickly, so I'd be out on the roof cleaning the panels all the time if I were to have any chance of breaking even.
So my thought was that some enterprising company should buy up a few acres of land (or rooftops), and let individual homeowners sponsor small batches of solar panels, like 5kw or 10kw, in exchange for some sort of credit on their electric bill. A system like this would dramatically reduce the barriers to entry for individuals who'd like to pay for solar power, as well as vastly increase the economies of scale. Does any system like this currently exist?
Except Google lists my house in a different city and zip code than the one it's actually in.
Naturally, that information is not editable. I've complained about this to my Google employee friends for years (not in a whiney sort of way), and even they have no idea how to correct it. On the other hand, I suppose I can live with being placed in Ventura, CA rather than Carpinteria, CA.:-)
I don't run Folding@Home on my PS3 for two reasons. First, because their boneheaded screensaver locks out AutoPlay on CD's! My PS3 doubles as a CD/DVD/Blu-Ray player for my living room, and it's ridiculous to have to turn on my TV to navigate the menus whenever I pop in a CD; I just want the damn thing to start playing. Second, it would be nice if Folding had the option to automatically run at night only, so it would only use power at off-peak hours, and incidentally help heat my house at night.:-) I get cheaper electricity at off-peak hours anyway, and the PS3 hoovers electrons like they're going out of style. (SonyStyle?)
Pong. Multi-touch Pong. Two players, on one screen. No question about it.
Also:
1. The ability to delete a screenful of emails at once.
2. Bluetooth headset battery level display in menu bar.
3. Ability to turn off EDGE roaming. (This will be in 1.1.1, so I hear.)
4. Higher resolution!! 160dpi is just enough to make me wish it were 240dpi..
5. Voice-dialing, 3G, GPS, 32GB, WHUXGA, and WTFOMGBBQ!
Slashdot Mirror
That would be *slightly* inconvenient to wear, but think of the size of her left arm after a few months of wearing it?
Forget the ring -- give her a platinum-iridium bra.
The sooner we wipe out imperial units the better
.1337.
Just don't tell today's kids that the number of cubic feet in a gallon is
It's "aluminium". Get used to it.
So you're saying it's also "tantalium", "molybdenium", "platinium", and "lanthanium"? Get over it.
Good analysis. But what happens when the distribution of envelope amounts is unbounded? Note that with your geometric distribution, the expected gain for switching is still 25%, UNLESS you're at one endpoint of the distribution or the other. So if you spread out the distribution to make the endpoints infinitely unlikely to be reached, does that change anything?
I should have stated the problem more carefully; the amount in the smaller envelope is also a random variable. (E.g., you don't know if the envelopes contain $0.01 and $0.02, or $1 and $2, or $437 and $874, or $1 trillion and $2 trillion.) All you know is that you're either doubling your money by switching, or halving it, with equal probability. So why is the 125% expected return calculation (1/2 (0.5) + 1/2 (2)) not valid? Clearly if you knew in advance that the two envelopes contained exactly $5 and $10, your argument would work. (Because the larger the amount in your envelope, the less chance the other envelope has more.) But when the amounts in both envelopes are infinitely variable, except for the known 2:1 ratio, the answer is a bit more subtle. For example, suppose you peek in your original envelope and it contains $100. You don't have any prior information about whether the other envelope contains $50 or $200. So why wouldn't you switch? Does peeking in the first envelope actually give you any useful information? The puzzle is not as obvious as it seems.
Yes, but you're missing the REAL puzzle, which is that even the FIRST switch (with the calculated 1.25x expected return) doesn't gain any information. By symmetry, the expected return on the initial switch MUST be exactly 1.0x, yet the simple math says 1.25x. Where does the math/logic go wrong?
Here's an even better problem:
Suppose Monty Hall gives you a choice of two envelopes. Each envelope contains a check, and one of them is written for TWICE the amount of the other. So you pick an envelope.
Now, Monty gives you the chance to switch envelopes. (Assume Monty always gives you the chance to switch.) Logically, since your envelope contains X, the other envelope can contain either 0.5X or 2X, with 50% probability... So the expected value of switching envelopes is 50% (0.5X + 2X), or 1.25X. So, you should switch.
But here's the tricky part: Monty now gives you the chance to switch back! Since your new envelope contains Y, then by the same logic as above, the expected value of switching back is 1.25Y... So you should switch back. Right?
Clearly, something is wrong with this chain of thinking. Can you figure out what it is?
In the latter scenario, it may be our goal simply to preserve our initial odds. If so, it pays to toss a coin on the second choice. This way, quite regardless of the host's strategy, we will have our odds at 1/3 or above.
No, your odds with flipping the coin to choose between two doors will always be exactly 1/2, regardless of the host's strategy. (Think about it.) But if you know something about the host's strategy, then a non-random choice can give you better odds; 2/3 from switching in the classic version.
However the 6-bit screen only puts out 63+63+63+1 = 190 different colors.
It's technically still 64 + 64 + 64 = 192 different colors. That's because the contrast ratio is finite; the "black" setting yields dark red, dark green, and dark blue colors on the subpixels, respectively.
Crystal Sushi
I wrote an article on my blog with some related thoughts about solar. In particular, I've considered installing solar panels on my roof, but my geographical region has a tendency to accumulate dust very quickly, so I'd be out on the roof cleaning the panels all the time if I were to have any chance of breaking even.
So my thought was that some enterprising company should buy up a few acres of land (or rooftops), and let individual homeowners sponsor small batches of solar panels, like 5kw or 10kw, in exchange for some sort of credit on their electric bill. A system like this would dramatically reduce the barriers to entry for individuals who'd like to pay for solar power, as well as vastly increase the economies of scale. Does any system like this currently exist?
The best fliers, and there are not many
Darwin in action.
Except Google lists my house in a different city and zip code than the one it's actually in.
:-)
Naturally, that information is not editable. I've complained about this to my Google employee friends for years (not in a whiney sort of way), and even they have no idea how to correct it. On the other hand, I suppose I can live with being placed in Ventura, CA rather than Carpinteria, CA.
Go here.
In memory of them, I'm officially nicknaming Leopard: Vistapard.
Leoptard.
I've told you once, I've told you 1,073,741,824 times: don't approximate!!!
I don't run Folding@Home on my PS3 for two reasons. First, because their boneheaded screensaver locks out AutoPlay on CD's! My PS3 doubles as a CD/DVD/Blu-Ray player for my living room, and it's ridiculous to have to turn on my TV to navigate the menus whenever I pop in a CD; I just want the damn thing to start playing. Second, it would be nice if Folding had the option to automatically run at night only, so it would only use power at off-peak hours, and incidentally help heat my house at night. :-) I get cheaper electricity at off-peak hours anyway, and the PS3 hoovers electrons like they're going out of style. (SonyStyle?)
There's a simple solution to that - if you don't exceed the speed limit, they won't make any money from the cameras.
No, they will simply keep lowering the speed limit until people begin breaking it again.
Here's another interesting approach to air safety:
Helical Flight Logic
If brute force isn't working... you aren't using enough of it.
I'm pretty sure that $1 doesn't go very far on yacht purchases.
Maybe he's purchasing a yacht nano.
Pong. Multi-touch Pong. Two players, on one screen. No question about it.
Also:
1. The ability to delete a screenful of emails at once.
2. Bluetooth headset battery level display in menu bar.
3. Ability to turn off EDGE roaming. (This will be in 1.1.1, so I hear.)
4. Higher resolution!! 160dpi is just enough to make me wish it were 240dpi..
5. Voice-dialing, 3G, GPS, 32GB, WHUXGA, and WTFOMGBBQ!
-Ben
...Scaled Composites isn't a government contractor.
Well, no, but it was just acquired by one.