Monty Hall, we have a PROBLEM

[Twitch Boy]

Re: Monty Hall, we have a PROBLEM

The tricky part is that you don't know whether the odd coin is heavier or lighter than all the others.

Okay, yup, that's what I didn't catch. Your last weighing would have to be between (at maximum) a known real coin on side A, an unknown coin on side B, and an unknown coin set aside. If they don't balance, the unknown coin on the scale is the counterfeit; if they do, it's the unknown coin set aside.

 

[unofan]

Re: Monty Hall, we have a PROBLEM

Yes, but "can you work out all the permutations?" is the question. I doubt starting with 3 vs 3 would work. Even if you start with 4 vs 4 as Papa Baer suggests, there is still a subtle twist required on the second or third step in order to get the solution in only two more tries.

e.g, "start with 4 vs 4" if it balances, then....

If it does not balance, then....

The tricky part is that you don't know whether the odd coin is heavier or lighter than all the others.

the tricky part is step 2 if the two sides are unbalanced after starting 4v4. Took me forever to figure out how to divide those eight into groups of 3, 3, and 2 so that, no matter what, you can always figure it out on step 3.

 

[owslachief]

Re: Monty Hall, we have a PROBLEM

... there is still a subtle twist required on the second or third step in order to get the solution in only two more tries.

I believe I know what the subtle twist is (spoiler -->) Involves switching out a number of coins from one side after the first weighing if unbalanced. A key piece is to realize that if you narrow it down to four coins, it takes two measurements to determine the odd coin. If you have 3 coins and know that the lightest coin is counterfeit, it takes one step (also if you know the heaviest coin is counterfeit, it takes one step).

So yes start by doing a 4 on 4:

Notation: Let A, B, C, ... A1, A2 ... m be decision nodes for what to weigh next. (1), (2) and (3) are nth weighing in a path. {S1} | {S2} is a weighing with set S1 on the left pan, S2 on the right. < denotes the left side is lighter, > denotes heavier, and = means balanced. Let 1, 2, 3, ... 12 be arbitrarily numbered coins. The "odd coin" is the one that weighs differently from the other 11, who all weigh the same.

Start with four coins weighed against four others.
(1) {1, 2, 3, 4} | {5, 6, 7, 8}
A: {1, 2, 3, 4} < {5, 6, 7, 8}. The odd coin must be from among these 8.
B: {1, 2, 3, 4} > {5, 6, 7, 8}. The odd coin must be from among these 8.
C: {1, 2, 3, 4} = {5, 6, 7, 8}. The odd coin is somewhere in {9, 10, 11, 12}
A -> switch 3 coins from S2 to S1, and bring in 3 unweighed coins to S2:
(2) {1, 6, 7, 8} | {6, 9, 10, 11}
A1: {1, 6, 7, 8} < {6, 9, 10, 11}
A2: {1, 6, 7, 8} > {6, 9, 10, 11}
A3: {1, 6, 7, 8} = {6, 9, 10, 11}
A1 -> the odd coin is either 1 or 5, since switching out 3 coins from both sides didn't make a difference.
A11: (3) Weigh either 1 or 5 against a known non-odd coin. {1} | {9}. If {1} <> {9} the odd coin is 1. If not, it must be 5 (done).
A2 -> the odd coin is the heavier one and must be in {6, 7, 8}, since moving them to the left made the left side heavier.
A21: (3) {6} | {7}. If {6} < {7}, the odd coin is 7. If {6} > {7} the odd coin is 6. if {6} = {7}, the odd coin is 8. (done)
A3 -> the odd coin is one of the ones thrown out for the second weighing (it's in {2, 3, 4}. Also, the odd coin is lighter than the rest.
A31: (3) {2} | {3}. If {2} < {3}, the odd coin is 2. If {2} > {3} the odd coin is 3. if {2} = {3}, the odd coin is 4. (done)

B -> this branch asks us to do everything the same as in branch A, but to reverse all the inequality signs.

C -> the odd coin is in {9, 10, 11, 12}. There is only one path to follow from here ->
C1: Make two weighings
(2) {9} | {10} and
(3) {9} | {11}.
If {9} <> {10} and {9} <> {11}, the odd coin is 9.
If {9} <> {10} and {9} = {11}, the odd coin is 10.
If {9} = {10} and {9} <> {11}, the odd coin is 11.
Finally, if {9} = {10} and {9} = {12}, the odd coin is 12.

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

This is not only a brain teaser, it's a real-life situation as well.

We have a very large pine tree at the back edge of our lot. A branch from it has grown out so that it is above a smaller tree and some ornamental shrubs. Spouse wants the overhang removed in case it gets full of snow and falls atop said smaller tree and shrubs. She and son discussed a plan in which we'd tie a rope to the 18' of branch, loop it over a higher branch, then saw the extending limb with a chain saw on an extension pole. They wanted to make sure that the weight of the falling limb wouldn't damage the tree or shrubs below.

They were both out running errands today, and so I was able to remove the tree limb quite easily, without any damage to the plantings below, and without using any rope. Solution in white below.

It didn't start out as a puzzle. I looked at the limb and recognized that after I cut it down, I'd trim the branches and cut the main stem into 12" lengths. Then I realized that I could cut the branches and trim the limb ahead of time instead of afterward. So I cut the branches off in place, and pile them up. each of them fell harmlesly onto the shrubs below, landing gently in the branches. Then I cut off the limb in 12" segments, each of which bounced harmlessly off the shrubs to the ground.

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

While the setup is humorous, this actually is a serious question as well.

If the opposite of inhale is exhale, and the opposite of interior is exterior, what is the opposite of increment?


Hint in white below:

Increase : decrease : : increment : ?

 

[Patman]

Dumb question for you math geeks who have time and effort (and motivation, which I have none):

Using a 1-16 confidence factor, as some sports bars use, put a "1" on the team you think is going to win but have no confidence in, and "16" on the team you think WILL win hands down (Example: 1 on MN vs CAR this past week, 16 on DEN over JAX).

How's the math on betting $10 on your "1," increase bet in $10 increments up to $160 for your "16?" Profitable strategy in the long run? Money line only, no spread bets.

Its really about how much noise you are willing to admit.

I've often mentioned that these things are as much game theory as probability theory.

If the confidence factor goes + or - on result then as long as your picks have better than half odds then the expected value is always maximized when you bet with highest confidence all the time.

If its one of those where you need to top whatever (say 3) of 50 people, a high risk strategy is most likely to get you there despite the downside. Same idea as pulling the goalie, you need to up your risk to make the difficult more likely... But you may also put the puck in your own net. But playing conservative gets you nothing.

If you build up a strong lead then you can start to hedge lower as you become more concerned about yielding ground. Just like Jeopardy when the wager is zero.

If you need to use all values 16 through 1, it does make life a bit more difficult but it does impose a constraint that you can find a solution that minimizes noise/variation. Mathematically, the solution is likely along the lines of going from more or less likely.

Edit: a lot of you example deals with the value of the result. If any choice is a guaranteed loser (as in you WILL lose money given your accepted probability) then you have to give it a 1.

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

Let's say you are an eccentric multi-millionaire who wants to incentivize people to reduce their gasoline use. You will help people trade in lower mileage vehicles for higher mileage vehicles.

Two people apply to you for help. Each of them drives the same distance.

One of them wants to trade up a pickup truck that gets 12 miles per gallon for one that gets 14 miles per gallon.

The other wants to trade up a compact car that gets 30 miles per gallon for one that gets 42 miles per gallon.

Which trade will save more gasoline?


What does this answer suggest about our current metric?

 

[JF_Gophers]

Let's say you are an eccentric multi-millionaire who wants to incentivize people to reduce their gasoline use. You will help people trade in lower mileage vehicles for higher mileage vehicles.

Two people apply to you for help. Each of them drives the same distance.

One of them wants to trade up a pickup truck that gets 12 miles per gallon for one that gets 14 miles per gallon.

The other wants to trade up a compact car that gets 30 miles per gallon for one that gets 42 miles per gallon.

Which trade will save more gasoline?


What does this answer suggest about our current metric?

The 14 MPG saves more if your look at % of a gallon used per single mile. but only slightly.

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

The 14 MPG saves more if your look at % of a gallon used per single mile. but only slightly.

More than "only slightly" it seems to me. If each drives 840 miles, then 12 --> 14 saves 10 gallons (70 --> 60) while 30 --> 42 saves 8 gallons (28 --> 20). That's 25% more savings for the former compared to the latter!



A much more useful measure than "miles per gallon" is "gallons per 100 miles." Then you can directly compare how much gasoline a car would use when you think about how far you drive, and you could more easily quantify how much it would cost you to fuel it.

 

[St. Clown]

Re: Monty Hall, we have a PROBLEM

Let's say you are an eccentric multi-millionaire who wants to incentivize people to reduce their gasoline use. You will help people trade in lower mileage vehicles for higher mileage vehicles.

Two people apply to you for help. Each of them drives the same distance.

One of them wants to trade up a pickup truck that gets 12 miles per gallon for one that gets 14 miles per gallon.

The other wants to trade up a compact car that gets 30 miles per gallon for one that gets 42 miles per gallon.

Which trade will save more gasoline?


What does this answer suggest about our current metric?

There are a whole slew of metrics involved here not being considered. How many miles does each vehicl drive each year? To what purpose are they being put? Is one a pleasure vehicle while the other's for business? Statistically speaking, if they're both family vehicles, we know that people who buy more efficient vehicles also tend to driver greater distances during a given year. Are either of them using the vehicle to start a new business? Do we value one over the other? If I was giving away vehicles, or just the money to purchase the vehicle, I'd have many more questions to answer other than simple EPA efficiency gains.

 

[JF_Gophers]

More than "only slightly" it seems to me. If each drives 840 miles, then 12 --> 14 saves 10 gallons (70 --> 60) while 30 --> 42 saves 8 gallons (28 --> 20). That's 25% more savings for the former compared to the latter!



A much more useful measure than "miles per gallon" is "gallons per 100 miles." Then you can directly compare how much gasoline a car would use when you think about how far you drive, and you could more easily quantify how much it would cost you to fuel it.

well it seems slight when using small decimals! The 42 car would have to get over 46 to become an equal or better savings.

 

[LynahFan]

Re: Monty Hall, we have a PROBLEM

A much more useful measure than "miles per gallon" is "gallons per 100 miles." Then you can directly compare how much gasoline a car would use when you think about how far you drive, and you could more easily quantify how much it would cost you to fuel it.

Europeans often quote efficiency in "liters per hundred km," just as you suggest.

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

There are a whole slew of metrics involved here not being considered. How many miles does each vehicl drive each year? To what purpose are they being put? Is one a pleasure vehicle while the other's for business? Statistically speaking, if they're both family vehicles, we know that people who buy more efficient vehicles also tend to driver greater distances during a given year. Are either of them using the vehicle to start a new business? Do we value one over the other? If I was giving away vehicles, or just the money to purchase the vehicle, I'd have many more questions to answer other than simple EPA efficiency gains.

Each of them drives the same distance [emphasis added]

Can we re-phrase that to be "all else being equal"?

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

I found my old quantum mechanics notes from college when I was doing some reorganizing.

This was a classic: "You perfectly balance a pencil standing upright on its point. How long until it falls over?"

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

Here is a thermodynamics question for the experts out there.

I want to heat water using two solar heaters. Each of them is 66' of black plastic pipe, 1-1/2" diameter. The pipe coils around around a cone and is then covered with a clear plastic dome to trap the heat.

I want to use the output from the pair of heaters to warm water in a swimming pool.

My conundrum is that I'm not sure which way to connect the two heaters (series or parallel) or whether it even makes a difference (if the warming function is non-linear then it does make a difference, depending upon which way the inflection of the curve bends, if the warming function is linear then it shouldn't matter).

Am I "better off" attaching the heaters in line, so that the first heater warms the water somewhat and then the second heater warms the same water even more? While the output water is warmer, I am only heating half as much water at a time.

Or am I "better off" splitting the water flow so that it travels equally through both heaters at the same time? While the output water is not as warm, I am at least warming twice as much water at a time.


Like I said, if Function(heat input) --> (increase in warmth) output is a linear relationship, then it shouldn't matter which way I choose.

Assume I connect in line for the following analysis:
If the F(heat) --> warming of the second in-line heater increases faster than the F(heat) --> warming of the first heater, then I am better off connecting them in line.
BUT
If the F(heat) --> warming of the second in-line heater increases more slowly than the F(heat) --> warming of the first heater, then I am better off connecting them in parallel.

Anyone with knowledge and experience who can weigh in here?

PS if it matters, each heater holds just over 6 gallons of water at any moment, if I did my math correctly.

pi*(.75")^2 * 66' = .8 cu ft = 6 gallons.

I suppose another consideration might be whether the pump has enough horsepower to push the water through 132' in line compared to 66' in parallell......

 

[owslachief]

Re: Monty Hall, we have a PROBLEM

Could this be analogous to an electrical circuit?

P = IR².

 

[LynahFan]

Re: Monty Hall, we have a PROBLEM

The total power you collect is just set by the size/shape/orientation/latitude of your device, which will be the same in both cases. Therefore, any difference in the rate of the temperature rise of your pool will be determined by the amount of power you *lose* out of the system - i.e. heat that leaks out in various ways. These would include conduction through your structure/supports, re-radiation from your collector, convection to the surrounding air, etc. All other things being equal, all of those mechanisms will allow more heat to leak out as the temperature of your collector increases. Therefore, your goal should be to keep the temperature as low as possible - seems a little counter-intuitive, but it's true. Therefore, you are better off running the two in parallel, so you get the same temperature rise across each one rather than adding the temperature rises together and creating a very "hot spot" at the outlet of the second unit. That will minimize the heat that leaks from the system and give you the faster temperature rise of the pool.

Unless, of course, running them in series means that your pump is working so hard that the extra energy it is putting into the system makes up for the additional heat loss...but heating the pool with electrical power is probably not what you had in mind!

Also, note that this is true regardless of whether the temperature rise of water with heat input is linear or not. As with most substances, the specific heat of water increases with temperature, so it's not linear - adding 100J of energy to cold water will result in a bigger temperature rise than adding the same 100J to warmer water. Regardless, though, you've still added the same amount of heat, so it doesn't matter. If everything were perfectly insulated (so that the heat leakage effect I mentioned before doesn't come into play), then you'd still be adding 100J to your pool, and the temperature would rise exactly the same amount.

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

Therefore, you are better off running the two in parallel.

Thank you for your thoughtful and well-reasoned response.

I had come to the same conclusion based on [apparently] different reasoning: by running the two in parallel, the rate of flow through each one would be reduced and so the water would stay in each heater longer, thereby imparting more heat intake per unit of water flowing through each heater.

 

[LynahFan]

Re: Monty Hall, we have a PROBLEM

Thank you for your thoughtful and well-reasoned response.

I had come to the same conclusion based on [apparently] different reasoning: by running the two in parallel, the rate of flow through each one would be reduced and so the water would stay in each heater longer, thereby imparting more heat intake per unit of water flowing through each heater.

No worries - pretty much what I do for a living (improving energy efficiency of aircraft).

Your reasoning is actually incorrect, though. Yes, if you circulate the water more slowly, the temperature difference (from the inlet temp to the outlet temp) will be higher so it may seem like "it heated up more." But, that higher temperature means more heat will be leaking out, so the net energy into the system will be lower. You actually still want a very high flowrate through each loop to keep the temperature *down*. Think of it this way: the sun is adding heat to the collector, and as the heat piles up, some of it will leak away. So, you don't want to let it build up there (at the collector) - you want to sweep that heat away immediately (with the flowing water) before it has a chance to leak away. That keeps the collector temperature down and keeps the heat flowing into the water instead of leaking to the environment. Of course, higher flowrates require bigger pumps, so you have to decide how much more $$$ (capital) and electrical power (operating cost) you are willing to sink into the pump for that additional efficiency gain at the collector.

 

[FreshFish]

Re: Monty Hall, we have a PROBLEM

higher flowrates require bigger pumps, so you have to decide how much more $$$ (capital) and electrical power (operating cost) you are willing to sink into the pump for that additional efficiency gain at the collector.

It turns out that our existing pump is more than enough to push water through the solar heaters.

There were several problems in implementation but finally got it together.

The heaters came in kits. Made overseas. So their dimensions don't quite fit US dimensions. I was able to use 1-1/4" flexible connectors with hose clamps.
The kits were designed for serial connection. I had to re-jigger the connections using 1-1/4" PVC pieces.
Our existing piping is 1-1/2" and fortunatly I found a 1-1/4" to 1-1/2" flexible connector with hose clamps for that junction.
I needed also to install a bypass line to reduce the system pressure. About 12 PSI before I started. Without the bypass line, running just the heaters in parallel, brought system pressure up to 18 PSI, adding the bypass line brought it back to 12 PSI. So I have two heaters in parallel along with a 1-1/4" pipe in parallel as well, receiving water from and feeding water back to 1-1/2" incumbent piping.

I bought a thermometer and we'll see how much temperature changes. It was in mid 60s at the outset.