Hello, I just wanted to briefly discuss the Bags of Marbles brain teaser in "Logic Puzzles".
I still think that no matter how you look at it, the probability -is- 1/2.
The number of possibilities was NOT the question that was asked. You chose one of the white bags, and picked out a white marble. One marble remains in this bag that you chose. If the random bag you had selected was bag A, then the remaining marble is white.
If it was Bag C, the remaining marble is black.
The explaination you gave is what I believe proves that the probability, not overall things that can actually happen, is a 1/2 chance that the next marble is white. You are right. We are not choosing bags. We are about to pick the other marble in the bag you hadn't picked up yet.
The facts are clear. You selected a white marble before choosing this next marble. One marble remains in this bag. It cannot be bag B.
Bag A- (removed)White - White = 1 white remaining
Bag C- (removed)White - Black = 1 black remaining
Therefore, whichever marble it originally was, mathematically, is irrelivant. The question was, in exact words: "What is the probability that the remaining marble is white?" So honestly, the way I see things, is that you're naming marbles in Bag A and therefore calling picking either one a whole new condition that makes the probability modified to 2/3.
What I am trying to say is, that was not the question. Color mattered, not which exact marble. If the bag randomly selected was Bag A, the remaining marble is White. If it was C, it is black.
This answers the given question, so the answer -is- 1/2. Or am I forgetting something? Can you elaborate?
Also, I am sorry if I'm posting this in the wrong section. I'm new to this place, and I have no idea where to go...:P
This site says 1 / 2
This site along with ours says 2 / 3
And welcome to the forum!
I have a definite opinion on this but will hold it for awhile. My method uses a tree structure. Let us see what others think.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.