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## #1 2012-11-27 13:46:44

jello!
Member
Registered: 2012-11-27
Posts: 1

### Bags of Marbles Flaw

"""The Puzzle: You have three bags, each containing two marbles. Bag A contains two white marbles, Bag B contains two black marbles, and Bag C contains one white marble and one black marble.

You pick a random bag and take out one marble.

It is a white marble.

What is the probability that the remaining marble from the same bag is also white?

The Solution . . .

2/3 (not 1/2)

You know that you do not have Bag B (two black marbles) so there are three possibilities

You chose Bag A, first white marble. The other marble will be white
You chose Bag A, second white marble. The other marble will be white
You chose Bag C, the white marble. The other marble will be black

So 2 out of 3 possibilities are white.

Why not 1/2? You are selecting marbles, not bags."""

Answer is really 1/2, not 2/3.  Note that you are picking a "random bag" not marbles.  So one bag has remaining white marble, one doesn't.  The other bag is not a possibility with 2 black marbles.  So, 1/2.

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## #2 2012-11-27 15:27:15

MathsIsFun
Registered: 2005-01-21
Posts: 7,630

### Re: Bags of Marbles Flaw

Just for fun I ran some random trials in a spreadsheet, 1000 at a time, and got: 0.69, 0.67, 0.66, 0.67, 0.69, 0.69, 0.66

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #3 2012-11-27 18:13:28

noyourrong
Member
Registered: 2012-11-27
Posts: 1

### Re: Bags of Marbles Flaw

jello! wrote:

Answer is really 1/2, not 2/3.  Note that you are picking a "random bag" not marbles.  So one bag has remaining white marble, one doesn't.  The other bag is not a possibility with 2 black marbles.  So, 1/2.

No, you're wrong. You pick a random bag, but you already know the result of the pick from your random bag. You've passed the random bag stage (thus why we can eliminate bag b as a possibility) and are already at picking a white marble. There are three white marbles, each with the same probability of being picked. 2/3 white marbles are with another white marble, therefore 2/3 of the time you'll draw a second white marble.

As usual these probabilities are easier to look at once you make the example absurd.

Now there are 1000 bags. 998 of them have two black marbles, 1 bag has two white marbles, 1 bag has one of each.

Even though the problem states we picked a random bag, if it also tells us we got a white marble out of it we know that it was one of two bags. For the sake of the problem the other 998 bags might as well never existed. We know that out of the 2000 marbles, we are only looking at 3. 2 of those are with each other, the last one is with a black marble. p=2/3

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## #4 2012-11-27 18:53:37

MathsIsFun
Registered: 2005-01-21
Posts: 7,630

### Re: Bags of Marbles Flaw

Good point about making an absurd example ... a useful tool.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #5 2012-11-28 15:34:59

noelevans
Member
Registered: 2012-07-20
Posts: 236

### Re: Bags of Marbles Flaw

It looks like to me that there are several ideas co-mingling here.

1)  If we know we have a white marble chosen already then we had either bag A or C.
One of the bags has left a while marble and the other a black marble.  So the probability
of choosing a bag and getting a white marble is 1/2.  The probability that the remaining
marble in the bag is white is the same as choosing one of the remaining bags and then
picking the marble out of it.  P(White)=P(choose bag A)*P(marble in A is white) = (1/2)*1=1/2.
P(Black)=P(choose bag C)*P(marble in C is black) = (1/2)*1 = 1/2.
2)  If we know that we have bag A or bag C, but haven't gotten a marble out yet, then the
probability of getting a white marble by choosing  bag A and then choosing a marble out
of it is  P(W) = (1/2)*1 = 1/2.  With bag C it is P(W) = (1/2)*(1/2) = 1/4 and
P(B) = (1/2)*(1/2) = 1/4.  So the total probability for P(W)= 1/2 + 1/4 = 3/4 and P(B)=1/4.

In 2) the probability of getting a white marble is three times that of getting a black one.  The ratio
of 3/4 to 1/4 is 1/3.  Perhaps this is where the 1/3 comes from.  But this 1/3 is not a probability,
it is a ratio of two probabilities.

To settle the dilemma perhaps we should average the 1/2 and 3/4 to get 5/8.
Or average 1/2 and 2/3 to get 7/12.  Or average 2/3 and 3/4 to get 17/24.

I was once told by someone deep into probabilities to "Always look closely at the SAMPLE
SPACES."  Typically if you change the sample space, then you change the probabilities.

The sample space in 1)  is {{W},{B}} knowing we have chosen a W out of bag A or C.
The sample space in 2)  is {{W,W},{W,B}} knowing we have one of these two bags to pick from.

In 1) we know  bag A or bag C was chosen since we also chose a white marble from one of them.
In 2) we just have that bag A or Bag C are our options but nothing has been chosen from them.

But I'm not sure this is the same question posed in an earlier thread for which there was a
name attached ( so-and-so paradox?).  Is this the same question?

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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## #6 2013-11-04 13:49:01

H. Poincare
Member
Registered: 2013-11-04
Posts: 1

### Re: Bags of Marbles Flaw

The answer to the question is 1/2 not 2/3.  The reason being is that the question never says that you put the white marble back after you pull it out.  Thus it is assumed that that white marble is not put back.  However, if the question was reworded to state that you put the white marble back into the bag you pulled it from, then the answer would indeed be 2/3.

Last edited by H. Poincare (2013-11-04 13:51:31)

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## #7 2013-11-05 14:06:07

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,005
Website

### Re: Bags of Marbles Flaw

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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