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bonanova
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The marbles in John's collection were of 13 different colors; each marble was of a single color.

His sisters, Jane and Joanne, especially liked the red ones, and asked if they might have one each.

Feeling generous, but not wanting to give them everything they asked, John let them each take

one marble from the black velvet bag where he kept his collection, but he did not let them look inside.

It is an even money bet that both sisters got their wish and pulled out a red marble.

How many red marbles might have been in John's collection before his sisters chose?

Edit: Of the 13 possible colors, obviously some were red. There may have been zero of some of the other 12 colors.

Edited by bonanova
clarify the color distribution possibilitites
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so there are 12 different colours (other than red). lets call this number x. So after both draws from the bag the number of red marbles must average at x for it to be an even bet. now we just have to find out the combinations as the maths is above.

Got Geography C/w to do so if its still unsolved then ill figure it out later

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In order for it be the same probability for both to pick a red marble, all the marbles, or none of them must be red, but seen as there are 13 different coloured marbles, there must be no red ones... right?

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  Asque said:
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In order for it be the same probability for both to pick a red marble, all the marbles, or none of them must be red, but seen as there are 13 different coloured marbles, there must be no red ones... right?

Nope.

The OP specifies just one probability, not two.

It is an even money bet that both sisters got their wish ....

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a bit more of the maths

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(x/y)*((x-1)*(y-1))=0.5

where x is the number of red marbles and y is the total number of marbles

edit: typo

Edited by reaymond
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(x/y)*((x-1)*(y-1))=0.5

where x is the number of red marbles and y is the total number of marbles

so:

(x/y)(x-1)(y-1) = 0.5

(x-1)(y-1) = xy-x-y+1

Multiply that by x/y and we have:

x2y - x2 - xy + x

all divided by y

= 0.5

subtract 0.5 from both sides

x2y - x2 - xy + x

-------------------------------------------------- - 0.5 = 0

y

split it up into different parts:

x2y/y - x2/y - xy/y + x/y - 0.5 = 0

x2 - x2/y - x + x/y - 0.5 = 0

combine some like terms

x2 - x + (x2 + x)/y - 0.5 = 0

x2 - x + (x+1)x/y - 0.5 = 0

divide all sides by x

x - 1 + (x+1)/y - 0.5/x = 0

0.5/x = (1/2)/x = 1/(2x)

so

x - 1 + (x+1)/y - 1/(2x) = 0

Now x and y must both be positive integers, and y is equal to or larger than 13, and x is equal to or larger than 1

x + (x+1)/y - 1/(2x) = 1

(x+1)/y - 1/(2x) = 1 - x

Because 1-x must be negative, 1/(2x) must be bigger than (x+1)/y

1/(2x) > (x+1)/y

make both denominators equal to 2xy:

y/(2xy) > (2x2+2x)/(2xy)

which means y is bigger than 2x2 + 2x

y > 2x(x+1)

...

Right now we have:

(x+1)/y - 1/(2x) = 1 - x

2x(x+1)/(2xy) - y/(2xy) = 1 - x

(2x(x+1) - y)/(2xy) = 1 - x

both sides of the equation are negative right now, but if we multiplied both by -1..

(y - 2x(x+1))/(2xy) = x - 1

Well I have to go, I'll think more about it later :P I know I didn't really do much lol, though I suspect the knowledge of 13 different colors must weed out certain solutions.... hmmm...

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  rossbeemer said:
You are most likely looking for a somewhat simple solution. However,

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John began with 10000000000 marbles, of which 7071067812 were red.

That's a lot of marbles.

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the OP doesn't atually say that there were exactly 13 marbles in the bag

Edited by lemonymelon
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  rossbeemer said:
You are most likely looking for a somewhat simple solution. However,
  Reveal hidden contents
John began with 10000000000 marbles, of which 7071067812 were red.
That's a lot of marbles.

That's a correct answer, ;) and you are the first solver.

Now let's introduce a sanity check by stipulating normal

sized marbles and the ability for John to hand the bag

to his sisters. B))

Are there other solutions?

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Alright, I've never liked finding integer solutions to problems, so I just put this into excel and started searching.

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85 red marbles, 35 of other colors

85/120*84/119=7140/14280=1/2

I think you can hold a bag of 120 marbles.

Not exactly elegant, but I don't think there are any smaller solutions. Can anyone find a non-brute force way to find that solution?

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  Chuck Rampart said:
Alright, I've never liked finding integer solutions to problems, so I just put this into excel and started searching.

  Reveal hidden contents

85 red marbles, 35 of other colors

85/120*84/119=7140/14280=1/2

I think you can hold a bag of 120 marbles.

Not exactly elegant, but I don't think there are any smaller solutions. Can anyone find a non-brute force way to find that solution?

Let me retry, scratching my old post on this topic. I'll make a new equation.

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r = red marbles

t = total marbles

(r/t) * (r-1)/(t-1) = 1/2

looks better as:

®/(t) * (r-1)/(t-1) = 1/2

(r*(r-1)) / (t*(t-1)) = 1/2

multiple both sides by 2t(t-1)

2r(r-1) = t(t-1)

or

2r(r-1) - t(t-1) = 0

Now we want to solve for one of the variables, with none of that variable on the other side of the equation

This would involve a mixture of factoring and taking the square root if possible, though I don't see how yet

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  Chuck Rampart said:
Alright, I've never liked finding integer solutions to problems, so I just put this into excel and started searching.
  Reveal hidden contents
85 red marbles, 35 of other colors

85/120*84/119=7140/14280=1/2

I think you can hold a bag of 120 marbles.

Not exactly elegant, but I don't think there are any smaller solutions. Can anyone find a non-brute force way to find that solution?

John can certainly pass that bag. Nice. ;)

Can anyone find an even smaller solution?

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  bonanova said:
John can certainly pass that bag. Nice. ;)

Can anyone find an even smaller solution?

Uh-oh. The next smallest solution is not a solution at all, because it only has 6 non-red marbles. Is there a trick to the wording I'm missing?

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how precise do you want the answer?

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10 red marbles and 14 total in bag = 0.495 chance

13 red marbles and 18 total in bag = 0.509 chance

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15 red marbles and 21 total in bag = even chance

edit: add smaller solutions

Edited by reaymond
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Let's say that for the non-red marbles, there is no restrictions on their color or their number.

I think it's clear now that 13 colors was a red herring [npi], and it may be affecting the search for a smaller solution.

The OP did not intend to say there were at least 13 non-red marbles, but under some interpretations it does say that.

So, now let the OP say ...

The bag contains John's marble collection.

Some of his marbles are red.

It's an even bet that if his sisters each draw one marble, they will both be red.

How many red marbles might there have been in the bag before the sisters chose theirs?

  Reveal hidden contents

15, 85 and 7071067812

Is there a solution smaller than 15 red marbles?

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  bonanova said:

The bag contains John's marble collection.

Some of his marbles are red.

It's an even bet that if his sisters each draw one marble, they will both be red.

How many red marbles might there have been in the bag before the sisters chose theirs?

  Reveal hidden contents

15, 85 and 7071067812

Is there a solution smaller than 15 red marbles?

  Reveal hidden contents
Let's say that for the non-red marbles, there is no restrictions on their color or their number.

I think it's clear now that 13 colors was a red herring [npi], and it may be affecting the search for a smaller solution.

The OP did not intend to say there were at least 13 non-red marbles, but under some interpretations it does say that.

So, now let the OP say ...

To me the OP clearly implied 13 different colors present in the bag.

For this new problem statement the minimal solution is:

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3 red marbles in a bag of total 4.

(3/4)*(2/3) = 1/2

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