A Fistful of Brain Teasers
Posted by: Don Atkinson on 13 November 2017
A Fistful of Brain Teasers
For those who are either non-British, or under the age of 65………. The UK used to have a brilliant system of currency referred to as “Pounds, Shillings and Pence”. Simplified to £ ״ s ״ d. No! Don’t ask me why the “Pence” symbol is a “d”, just learn it and remember it !
A £ comprised 20 Shillings and a Shilling comprised 12 Pence. Thus a £ comprised 240 Pence. I reckon that both Microsoft and Apple would have difficulty with these numbers in their spreadsheets, more so if we included Guineas, Crowns, Half-Crowns and Florins. However, I digress..............
The purpose of the explanation is to assist with the first two or three teasers that follow. So just to ensure a reasonable comprehension has been grasped…. ….. if each of three children has £3 − 7s − 9d, then collectively they have £10 − 3s − 3d Got the idea ? Good ! Just try 5 children, two each with £4 − 15s − 8d and three each with £3 − 3s − 4d. How much do they have between them ? (this isn’t the first brain teaser, just the basic introduction with some “homework”, the Teasers follow)
The probability of this box being the gold/silver one is 1/3.
It’s obviously not the one with two silver coins. The remaining two boxes present three scenarios that show a gold coin: gold/gold box left compartment, gold/gold box right compartment and gold/silver box left compartment. In two of these the other coin is gold only the last has a silver coin on the other side.
I think the illustration leads to the assumption that the left compartment has been opened. There are only two possibilities for this scenario, which would lead to a probability of 1/2.
N.B Left and right weren’t identified in the original information (and might be randomly reversed anyway!).
Hi IB,
that was very much the point I was trying to make. The left compartment/right compartment relates to Dons illustration. I could have said compartment a and b or something similar.
Innocent Bystander posted:N.B Left and right weren’t identified in the original information (and might be randomly reversed anyway!).
Left and right are irrelevant. The original information is straightforward. Apart from the picture, there is no mention of left or right. And if I had drawn the picture the other way around it wouldn’t make any difference to the question or the answer.
Don Atkinson posted:Innocent Bystander posted:N.B Left and right weren’t identified in the original information (and might be randomly reversed anyway!).
Left and right are irrelevant. The original information is straightforward. Apart from the picture, there is no mention of left or right. And if I had drawn the picture the other way around it wouldn’t make any difference to the question or the answer.
That was my point!
Innocent Bystander posted:Don Atkinson posted:Innocent Bystander posted:N.B Left and right weren’t identified in the original information (and might be randomly reversed anyway!).
Left and right are irrelevant. The original information is straightforward. Apart from the picture, there is no mention of left or right. And if I had drawn the picture the other way around it wouldn’t make any difference to the question or the answer.
That was my point!
Ah good !
so have you now figured out what the correct probability is ?
So did I get it wrong?
Hi HH,
I wasn’t implying whether IB had the right probability or the wrong probability. I was just asking him to clarify his position.
Cheers, Don
So was I right? And I didn’t even get help.....
1/2. Only two boxes have gold/silver coins
If the question asked you to open more boxes then it would be different, but the question doesn't. It's just asking the probability of it being either one or the other.
Don Atkinson posted:Hi HH,
I wasn’t implying whether IB had the right probability or the wrong probability. I was just asking him to clarify his position.
Cheers, Don
My position is unchanged: 1/2, 50:50 or however one wishes to express - the left/right bit was only in response to Mulberry’s introduction of the handing.
...which was only my attempt to say that there are three possible ways to see a gold coin. Under both lids of the gold/gold box and under one lid of the silver/gold one. Only one of these three has a silver coin on the other side -> 1/3 chance.
Mulberry posted:...which was only my attempt to say that there are three possible ways to see a gold coin. Under both lids of the gold/gold box and under one lid of the silver/gold one. Only one of these three has a silver coin on the other side -> 1/3 chance.
But as I see it, one, the silver-silver box, has already been excluded by the fact of the opened lid revealing a gold coin, and so is no longer in the equation, hence /2 not /3.
at Mullberry.
But, we're looking for a silver coin.
Innocent Bystander posted:Mulberry posted:...which was only my attempt to say that there are three possible ways to see a gold coin. Under both lids of the gold/gold box and under one lid of the silver/gold one. Only one of these three has a silver coin on the other side -> 1/3 chance.
But as I see it, one, the silver-silver box, has already been excluded by the fact of the opened lid revealing a gold coin, and so is no longer in the equation, hence /2 not /3.
Or to put it another way, the question is not what is the probability of the second lid of the same box revealing silver, which is what your response answers, but what is the probability that this half opened box is the one with the other half containing a silver coin. The consideration is of boxes, not of individual compartments, with already it known that this box must be either the gold-gold or the gold-silver.
Maybe 2/3
Since the question doesn't ask me to open any more boxes to determine the outcome. I'm assuming we know already what's in the boxes. I know one box doesn't and the opened box might and the other unopened box might as well.
hungryhalibut posted:The probability of the second coin being also gold is 2/3, therefore it being silver must be 1/3. It’s the opposite of the standard solution to the Bertrand’s box paradox, which is rather like that ‘do I switch to the other cup’ Monty Hall problem. My annoyingly good at maths son tries to explain it to us, invariably after we have had too much wine.
HH,
What you have stated in the first two sentences above is absolutely correct.
I will take in good faith that what you have said in your last sentence is also absolutely correct.
hungryhalibut posted:So did I get it wrong?
See my comment above
Super, thanks.
Toby.
Your last sentence should read or, not and.
Evens.
Don Atkinson posted:hungryhalibut posted:The probability of the second coin being also gold is 2/3, therefore it being silver must be 1/3. It’s the opposite of the standard solution to the Bertrand’s box paradox, which is rather like that ‘do I switch to the other cup’ Monty Hall problem. My annoyingly good at maths son tries to explain it to us, invariably after we have had too much wine.
HH,
What you have stated in the first two sentences above is absolutely correct.
I will take in good faith that what you have said in your last sentence is also absolutely correct.
Once the first gold coin is revealed. And it has been revealed. The chance of revealing a second gold is 1/2.
Innocent Bystander posted:[...] the question is not what is the probability of the second lid of the same box revealing silver [...] but what is the probability that this half opened box is the one with the other half containing a silver coin.
I think these are actually the same question. Only if the second lid reveals silver, the half-opened box is the one with the other half containing silver (to reassamble your post).
fatcat posted:Don Atkinson posted:hungryhalibut posted:The probability of the second coin being also gold is 2/3, therefore it being silver must be 1/3. It’s the opposite of the standard solution to the Bertrand’s box paradox, which is rather like that ‘do I switch to the other cup’ Monty Hall problem. My annoyingly good at maths son tries to explain it to us, invariably after we have had too much wine.
HH,
What you have stated in the first two sentences above is absolutely correct.
I will take in good faith that what you have said in your last sentence is also absolutely correct.
Once the first gold coin is revealed. And it has been revealed. The chance of revealing a second gold is 1/2.
You might think that, but it’s not. It’s 2/3. Therefore the probability of it being silver is 1/3, as the probabilities sum to 1.
fatcat posted:Once the first gold coin is revealed. And it has been revealed. [...]
But the first gold is twice as likely to be in the gold/gold box. If you look at the Illustration and pick one of the gold coins, what is the other coin in that box? For both (=2) in the g/g box it's gold, for the one (=1) in the g/s box it's silver. 2+1=3 outcomes, only one with silver -> 1/3.
If I open either lid on one box it could be silver. The unopened lid on the opened box could be silver and/or either lid on the other box could be silver... is that right ?