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Math Help - Probability: Red or Blue

  1. #1
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    Probability: Red or Blue

    In a box, I have a ball, which has an unknown colour (can be either red or blue). I put in another ball, which is red, shake the box and take out a random ball. The ball taken out turns out to be red. If I take out the another ball, what is the chance of drawing a red ball?

    Right after I read that question, I'm thinking the answer is 50%. The problem is, I have to have some sort of written proof. Can someone help me with that?
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  2. #2
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    Hello, Grey!

    The answer of 50% is much too easy.
    . . It is a classic trick question.


    In a box, I have a ball, which has an unknown colour (can be either red or blue).
    I put in another ball, which is red, shake the box and take out a random ball.
    The ball taken out turns out to be red.
    If I take out the another ball, what is the chance of drawing a red ball?
    There are four equally likely outcomes.

    (A) The box contains a red ball, call it R1.
    . . .A red ball is added, call it R2.

    . . .(1) We draw R1, then we draw R2.
    . . .(2) We draw R2, then we draw R1.


    (B) The box contains a blue ball, call it B1.
    . . .A red ball is added, call it R2.

    . . .(3) We draw R2, then we draw B1.
    . . .(4) We draw B1, then we draw R2.


    We are told that the first ball is red.
    . . Hence, case (4) is discarded.

    We have three equally likely outcomes:
    . . (1) We draw R1, then we draw R2.
    . . (2) We draw R2, then we draw R1.
    . . (3) We draw R2, then we draw B1.

    In two of the three outcomes, the second ball is red.

    Therefore: .P(second ball is red) .= .2/3

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    This is identical to another classic.

    I know my neighbor has two children.
    I meet one of them and the child is a girl.
    What is the probability that the other child is a girl?

    The "obvious" answer is 50%; the child must be a boy or a girl, right?
    . . Wrong!


    There are three possible cases.

    He has one girl, one boy.
    . . .(1) You met the girl; the other is a boy.

    He has two girls: G1 and G2.
    . . (2) You met G1; the other is G2.
    . . (3) You met G2; the other is G1.

    In two of the three cases, the other child is a girl.

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  3. #3
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    Oh, I should have seen that. My mind went instantly to 50% when I first read the question. Thanks
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