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Topic: Gold or Silver? (Read 2646 times) |
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ThudnBlunder
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Gold or Silver?
« on: Jun 10th, 2003, 10:58am » |
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[Not sure if this has been posted before.] 
There are two boxes, each containing two coins.
One box contains a Gold coin and a Silver coin.
The other box contains two Silver coins.
A Silver coin is drawn from a box that is chosen at random.
What is the probability that the other coin in the box is also Silver?
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| « Last Edit: Jun 10th, 2003, 11:00am by ThudnBlunder » |
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wowbagger
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Re: Gold or Silver?
« Reply #1 on: Jun 10th, 2003, 11:27am » |
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50%, because you chose the box, not a particular coin.
This is a little like the two-child family riddles in the medium section, isn't it?
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towr
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Re: Gold or Silver?
« Reply #2 on: Jun 10th, 2003, 11:33am » |
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Well, it depends, if the drawn silver coin is the result of the deliberate drawing of a silver coin, or the drawing of a random coin from the box. If the latter is the case than the answer may be different from 50%.
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| « Last Edit: Jun 10th, 2003, 11:36am by towr » |
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wowbagger
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Re: Gold or Silver?
« Reply #3 on: Jun 10th, 2003, 11:37am » |
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It says
on Jun 10th, 2003, 10:58am, THUDandBLUNDER wrote:| A Silver coin is drawn from a box that is chosen at random. |
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To me, this means that the box is chosen at random. After that, it doesn't matter which silver coin you take if you chose the box with two silver coins.
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ThudnBlunder
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Re: Gold or Silver?
« Reply #4 on: Jun 10th, 2003, 11:40am » |
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Quote:| This is a little like the two-child family riddles in the medium section, isn't it? |
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Yes, it is a little like them.
Quote:| Well, it depends, if the drawn silver coin is the result of the deliberate drawing of a silver coin, or the drawing of a random coin from the box. |
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A silver coin is drawn at random.
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wowbagger
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Re: Gold or Silver?
« Reply #5 on: Jun 10th, 2003, 11:42am » |
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on Jun 10th, 2003, 11:40am, THUDandBLUNDER wrote:| A silver coin is drawn at random. |
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If you want to choose the coin (rather than the box) at random, why don't you say so in your original post? 
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MattyDK23
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Re: Gold or Silver?
« Reply #6 on: Jun 10th, 2003, 11:51am » |
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The way I read it...
You chose a box at random, and pick a coin. You only look at cases where the first coin is silver.
In that case, the probability is 66% (or 2/3), isn't it? I seem to remember that from my stats class two years ago.
I just sketched up a really crude C program that ran 10,000 iterations...it came up with roughly the same probability.
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towr
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Re: Gold or Silver?
« Reply #7 on: Jun 10th, 2003, 11:52am » |
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box1 silver
box1 gold
box2 silver
box2 silver
so if you drew a silver coin 2 to 3 it's box 2 with another silver.
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wowbagger
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Re: Gold or Silver?
« Reply #8 on: Jun 10th, 2003, 11:57am » |
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As (almost) always, it's a question of how you model the choosing and drawing process.
Matty, do you try to "choose" a silver coin from the box with the gold coin in it?
Towr, I completely agree with your last post. However, I choose a box first, so the second coin is silver in only 50% of the cases.
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MattyDK23
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Re: Gold or Silver?
« Reply #9 on: Jun 10th, 2003, 12:14pm » |
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on Jun 10th, 2003, 11:57am, wowbagger wrote:| Matty, do you try to "choose" a silver coin from the box with the gold coin in it? |
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The selection of the first silver coin is an assumption in the 'riddle' / question, at least in the way I read it. We're past the first stage, past the part where the first coin has already been chosen.
In my program, I have two 'stages'. Everything has equal probability -- picking a random coin from a random box. However, I only recorded the occurences of
1. The number of times the first coin was silver
2. The number of times both the first and second coin were silver
My output was:
Total Iterations: 10000
First Coin Was Silver: 7503 times
Both Were Silver: 5054 times
Conditional Probability: 5054/7503 = 0.6736
So, there was 10000 - 7503 = 2497 times that the first coin was gold. And 7503 - 5054 = 2449 times that the second coin was gold when the first coin was silver.
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| « Last Edit: Jun 10th, 2003, 12:17pm by MattyDK23 » |
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towr
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Re: Gold or Silver?
« Reply #10 on: Jun 10th, 2003, 12:18pm » |
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on Jun 10th, 2003, 11:57am, wowbagger wrote:
Towr, I completely agree with your last post. However, I choose a box first, so the second coin is silver in only 50% of the cases. |
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But you don't know which of the two boxes (identified by content) you choose.
Suppose you have two boxes, one with a million silver coins, and one with 999,999 gold coins and one silver coin. You pick a box, you randomly pull out a coin from 'your' box, and it happens to be silver. What are the chances you have the all silver box?
It's about knowledge. In a way you're right, but the extra knowledge would put the bayesian guess elsewhere.
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| « Last Edit: Jun 10th, 2003, 12:18pm by towr » |
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ThudnBlunder
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Re: Gold or Silver?
« Reply #11 on: Jun 10th, 2003, 12:20pm » |
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Quote:| If you want to choose the coin (rather than the box) at random, why don't you say so in your original post? |
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OK, a Silver coin is chosen at random from a box that is chosen at random .
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TenaliRaman
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Re: Gold or Silver?
« Reply #12 on: Jun 10th, 2003, 12:21pm » |
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Does the knowledge that we have drawn one silver coin affect the probability in any way??
My guess it something like the Monty Hall Problem kinda thing!
So the question is (As i see it)
what is the probability that we choose a box containing 2 silver coins?
obviously i feel the answer is 50%
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MattyDK23
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Re: Gold or Silver?
« Reply #13 on: Jun 10th, 2003, 12:27pm » |
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Yeah, sorry...I said my part kinda wrong.  You still have to keep in mind the differences between the two boxes. I kinda just overstated the part about the conditional / assuption part.
Just take what towr said, and multiply it by 2. 
Anyways, just look at the probabilities that the program spits out.
First coin silver: 75% of the time (0.75)
Both coins silver: 50% of the time (0.5)
Prob. of 2nd coin silver, given first coin is: 0.5 / 0.75 = 0.67
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wowbagger
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Re: Gold or Silver?
« Reply #14 on: Jun 10th, 2003, 12:30pm » |
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on Jun 10th, 2003, 12:18pm, towr wrote:
But you don't know which of the two boxes (identified by content) you choose.
Suppose you have two boxes, one with a million silver coins, and one with 999,999 gold coins and one silver coin. You pick a box, you randomly pull out a coin from 'your' box, and it happens to be silver. What are the chances you have the all silver box? |
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Well, after picking a box (with equal probability, I assume), you have the all silver box with 50%.
Quote:| It's about knowledge. In a way you're right, but the extra knowledge would put the bayesian guess elsewhere. |
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Yes, knowledge can be important.
I don't know nothing about that bayesian guess though.
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wowbagger
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Re: Gold or Silver?
« Reply #15 on: Jun 10th, 2003, 12:32pm » |
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on Jun 10th, 2003, 12:20pm, THUDandBLUNDER wrote:
OK, a Silver coin is chosen at random from a box that is chosen at random . |
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Hm, the box is chosen first, right? That's at least how I interpreted the original problem statement. So the probability of the second coin also being a silver coin has to be equal to the probability of choosing the all silver box in the first place.
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towr
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Re: Gold or Silver?
« Reply #16 on: Jun 10th, 2003, 12:35pm » |
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a different extreme case.
we have again two boxes, one with one silver coin, one with one gold coin. You select a box at random, and select a coin from that box at random, it is silver. What is the chance you had the box with only silver coins.
(to show it does matter, a lot, which coin is drawn, even if there is an apriori 50-50 chance of either box)
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wowbagger
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Re: Gold or Silver?
« Reply #17 on: Jun 10th, 2003, 12:39pm » |
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Of course you get a different result in your new extreme case, towr. Now the box with the gold coin isn't a valid result anymore (due to the extra knowledge, as you rightly point out), whereas in the original puzzle it is!
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towr
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Re: Gold or Silver?
« Reply #18 on: Jun 10th, 2003, 12:47pm » |
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actually, you get the exact same kind of information there as well, only less of it.
suppose you were to pick a random box, and then a random coin from the box.
you get
box1 silver
box1 gold
box2 silver1
box2 silver2
The only difference in the original puzzle is that the second occurence isn't a valid option any more.
In my extreme case you have
box1 gold
box2 silver
The parrallel is that one option from the 'pick a random coin' case is invalidated because you know part of the endresult. So instead of 1 out of 2 it goes to 1 out of 1 in my extrene case. And from 2 out of 4, to 2 out of 3 in the original puzzle.
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| « Last Edit: Jun 10th, 2003, 12:49pm by towr » |
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towr
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Re: Gold or Silver?
« Reply #19 on: Jun 10th, 2003, 12:59pm » |
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Here's the Bayes view on it.
P(A|B) = P(A & B) / P(B) = P(B|A) * P(A)/P(B)
P(S) = Sum over all i of P(S | Ei) * P(Ei))
P(box1) = 50% (the a priori chance of box1)
P(box2) = 50%
P(silver | box1) = 50%
P(silver | box2) = 100%
P(silver) = 100% * 50% + 50% * 50% = 75% (like we didn't know that beforehand, the a priori chance of picking silver randomly from a box)
P(box1 | silver) = P(silver | box1) * P(box1) / P(silver) = 50% * 50% / 75% = 33% (1/3)
P(box2 | silver) = P(silver | box2) * P(box1) / P(silver) = 100% * 50% / 75% = 67% (2/3)
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Icarus
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Re: Gold or Silver?
« Reply #20 on: Jun 10th, 2003, 4:18pm » |
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You are all wrong! The probability that the other coin is silver is 100%.
Why? Because the guy drawing the coin is extremely greedy, and would have drawn the gold coin if it had been available! 
This is the same argument as in the two kids puzzles, and in the Monty Hall puzzle and in some others.
It turns out that the actual puzzle is determining how to interpret the statements in the puzzle.
In this one T&B tells us clearly that the box was chosen at random. He tells us nothing about how the coin was chosen from the box - only that the result was a silver coin.
What probability you get depends on what assumptions you make about how the coin was chosen.
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MattyDK23
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Re: Gold or Silver?
« Reply #21 on: Jun 10th, 2003, 5:01pm » |
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You've just expanded the question to include assumptions about the person's personality (greedy), the economy (value of gold vs. silver), etc...
One could also assume that the person was very modest...or that the value of gold had just plummeted...or that we're in an alternate universe where Gold is really Bronze, and vice versa...
We're working with what we're presented with...
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Icarus
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Re: Gold or Silver?
« Reply #22 on: Jun 10th, 2003, 8:45pm » |
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My point was that we were NOT presented enough to be able to calculate a probability.
However - in rereading the thread, I see that T&B has already addressed the problem by stating that the coin is selected at random from the box.
My apologies for scanning the thread too lightly, and reopening an already settled point.
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wowbagger
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Re: Gold or Silver?
« Reply #23 on: Jun 11th, 2003, 2:25am » |
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on Jun 10th, 2003, 4:18pm, Icarus wrote:It turns out that the actual puzzle is determining how to interpret the statements in the puzzle.
What probability you get depends on what assumptions you make about how the coin was chosen. |
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I'm glad you agree with me on this point.
Now for the finicky part:
on Jun 10th, 2003, 11:40am, THUDandBLUNDER wrote:
A silver coin is drawn at random.
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Is this really clear-cut? I could argue in favour of an interpretation like: "Of all silver coins, one is drawn at random." Can anybody relate to this?
On the other hand, it can mean: "A coin is drawn at random, and it turns out to be silver."
In this case we have to deal with conditional probability and arrive at towr's result.
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redPEPPER
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Re: Gold or Silver?
« Reply #24 on: Jun 11th, 2003, 2:45am » |
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on Jun 11th, 2003, 2:25am, wowbagger wrote:| I could argue in favour of an interpretation like: "Of all silver coins, one is drawn at random." Can anybody relate to this? |
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Yep. Although the other meaning is probably the intended meaning, the wording isn't very clear.
From T&B's update:
Quote:| OK, a Silver coin is chosen at random from a box that is chosen at random . |
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So, first, a box is chosen at random. That's 50% to pick the box with gold. In which case picking a silver coin at random will be easy, as there's only one siver coin. The other coin in that case will be gold. In the other 50% case, no matter what coin is picked, the other coin will be silver. In conclusion, the probability would be 50% that the other coin is silver.
A more accurate wording for the other meaning would be:
"A coin is chosen at random from a box that is chosen at random. It turns out it's a silver coin." And so on...
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