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Jul
22
answered 50/50 Joker of “Who wants to be a Millionaire” - A “Monty Hall Problem” variation?
Jul
21
comment Making 400k random choices from 400k samples seems to always end up with 63% distinct choices, why?
It might be helpful to add another couple steps to your derivation. The probability of each slot being marked by one item is 1/N, and the probability of not being marked by one item is 1-(1/N). The probability of each slot being marked by any of N independent items is thus (1-(1/N))^N, and since there are N slots with that probability, the expected total is thus N * (1-(1/N))^N.
Jul
18
comment Is value of $\pi = 4$?
BTW, on similar note, speaker whom I saw do the trick of cutting a huge hole in a small piece of paper (fold it and make alternating cuts on the inside and outside) gave what I thought was a brilliant explanation: when cutting a "normal" hole, attempts to reduce the surface are of paper required for each lineal inch of parameter also increases the amount of paper wasted in the hole itself and reduces the amount of paper available to fabricate the perimeter. The "trick" cutting technique allows the perimeter to be made thinner without increasing the loss of material.
Jul
18
comment Is value of $\pi = 4$?
When I saw the question, I was immediately reminded of the "diagonal" paradox is one of Dudney's puzzle books. I don't think I saw what I'd consider the "real" answer there nor here, however, which is that each time one subdivides the problem in half, the amount of error in each piece is cut in half, but the number of pieces doubles. In order for a limit computed via problem reduction to be valid, the per-piece error must decrease faster than the number of pieces increases.
Jul
18
comment Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?
To extend things a little further, draw a line segment which is tangent to the circle and leaves going "counter-clockwise", with the same length as AC. Since that line is tangent to the circle, its slope will match that of the circle at C. Translate that line segment so that the endpoint that was at C is now at A, and the other endpoint will be on the circle, 90 degrees counterclockwise from the original.
Jul
5
comment How to distinguish walking on a sphere or on a torus?
If one was on a sphere, by what means would you never know that circles one had constructed were geodesics?
Jul
5
comment How to distinguish walking on a sphere or on a torus?
If one can draw a ring, and then draw another ring which crosses the first ring only once, then one is on a torus. Failure to do that, however, does not imply that one is on a sphere. It could be that one hasn't drawn the rings in the right place.
Jul
5
comment How to distinguish walking on a sphere or on a torus?
...either classification would be deemed acceptable. Defining the smallest handle size of interest would make it possible to define things like the required size of triangles.
Jul
5
comment How to distinguish walking on a sphere or on a torus?
@robjohn: I wouldn't expect artificial structures to count as part of the planet, since otherwise buildings would count as "handles" unless all windows were sealed and each room could only be visited through one sequence of doors. On the other hand, tunnels would certainly alter the topology, whether they were truck-sized or mouse-sized. It's probably best to specify that a planet should be considered a "sphere" if it has no handles which are above a certain size, and a "torus" if it has handles which are above a certain (larger) size. If the largest handle is between the two sizes...
May
15
comment Why can't you add apples and oranges, but you can multiply and divide them?
@Jens: My point was that one could do math even without being able to reduce things, but multiplication by "compond" factors would cause expressions to quickly grow out of hand if one couldn't factor them. Perhaps a better example might have been an inductor in series with a capacitor. At any given frequency, conversions will exist from both henry and farad to reactive ohms, but without a specified frequency the addition must be processed using separate units.
May
14
comment Why can't you add apples and oranges, but you can multiply and divide them?
It may be worth noting that it is possible to add units, but the results aren't terribly useful. If one feeds (1 horsepower + 10 watts) worth of power into a system for one hour and 15 minutes, the total energy will be one horsepower hour plus 10 watt hours plus 15 horsepower minuts plus 150 watt minutes. Not the most convenient bunch of figures to work with.
Apr
25
comment Does 17% have to be equal to 0.17?
Actually, as a somewhat more interesting example, suppose someone said they reduced their mortgage rate by 1%, and someone else said they reduced theirs by 17%. Both people now have mortgages at a 5% rate. Would you guess the first person's old rate was about 5.05% or 6%? Would you guess the second person's old rate was about 6% or about 22%? My point is that the meaning of "%" can vary, so there's more to it than just a fractional numeric value.
Apr
25
comment Does 17% have to be equal to 0.17?
If the tax rate used to be 30%, saying it rose by "17%" would bean the new rate was about 35%. Saying it rose by ".17" could be read as implying a new rate of either "30.17%" or "47%". Saying "17 percentage points" would indicate the new rate was "47%".
Apr
21
comment If there are obvious things, why should we prove them?
@SteveJessop: That might be interesting to test. Of course, some people's ability to spot the obvious can be limited. One of my favorite examples of that is Eddie Kantar relating his efforts to teach people bridge; when asked where a certain card was after a sequence of plays, some people would reply that it must be in dummy, notwithstanding that all of dummy's cards were visible and the card in question was not among them!
Apr
21
comment If there are obvious things, why should we prove them?
@SteveJessop: If one specifies that the choice must be made in such fashion as to give the player no information, then the fact that the player gains no information should be obvious [it's stated in the puzzle!] If one specifies that if the player chooses #1, then host must show door #3 half the time when the prize is behind #1, and all the time when it's behind #2, then the fact that the prize is more likely to be in the location required for the "all the time" case is pretty clear. In any case, I think there are better examples of things that are "obvious but wrong".
Apr
21
comment If there are obvious things, why should we prove them?
...a player's expectation wouldn't go down by switching, but it might not go up either. For example, if after the player chooses #1 the host will alway show #3 unless it contains a prize (in which case he'll show #2) then having the host open door #3 would increase the odds a player's initial guess was right to 50-50, so the probability of a win from switching would also be 50-50. I've very seldom seen explicitly stated the fact that the host's choice must be random when it isn't forced, but it's essential for the claimed conclusion.
Apr
21
comment If there are obvious things, why should we prove them?
@SteveJessop: If a player selects door #1 and a host says "let's see what's behind door #3" and shows that door #3 is empty, it's not obvious (unless explicitly stated) what prompted that particular choice. Some ways of describing the host's method of selection would make it obvious that switching would win 2/3 of the time; others would be equivalent, but not instantly recognizable as such, and consequently would make the benefit of switching less obvious. Unless it is specified that in the "initial choice was correct" case the host will choose the other doors with equal probability...
Apr
21
comment If there are obvious things, why should we prove them?
@SteveJessop: For switching to be beneficial in the Monty Hall scenario, the probability that one would be offered a chance to switch if one is wrong must be at least half as high as the probability that would be offered a chance to switch if one was right. If someone looked at all 51 non-chosen cards and would only turn up 50 non-ace cards if the ace was the chosen card, and would otherwise turn up 50 cards including the ace ("sorry--you lose"), then switching would turn a 1/52 chance of winning into a 0% chance.
Apr
10
answered Which number is larger and Why? 1.7 or 1.73205
Apr
10
comment Which number is larger and Why? 1.7 or 1.73205
Rarely would 1.7 represent a uniform distribution over the interval 1.65 to 1.75; it would more typically represent a vaguely-bell-shaped distribution over a slightly larger interval (e.g. a physical attribute which is exactly 1.64 would be expected to read 1.6 most of the time, but sometimes read 1.7).