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Math Help - Conditional Probability and law of total probability.

  1. #1
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    Question Conditional Probability and law of total probability.

    This is a fun question i found on the internet, a bit harder than my course and ive spent hours on it but cant find a solution, i was hoping someone could help me.
    Here's the situation.
    You are in jail and have been sentenced to death tomorrow, however there's a way out.
    you're given 12 red balls and 12 blue balls with 2 urns, you have to fully distribute the balls between these two urns. the executioner will then select an urn and draw a ball
    if it is blue you live and if it is red you die.
    suppose you place k balls in to the first urn and 24-k in to the second
    suppose you put i blue balls in to the first urn and 12-i in to the second
    Let A be the event that you live.
    By conditioning on the urn that the executioner chooses, and using the law of total probability show that:

    P(A) = (12i - ki + 6k) / k(24-k)

    please could you explain the steps used so i can try and understand the solution and method.
    Thank you in advance
    Luke
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  2. #2
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    Re: Conditional Probability and law of total probability.

    Quote Originally Posted by luke95 View Post
    This is a fun question i found on the internet, a bit harder than my course and ive spent hours on it but cant find a solution, i was hoping someone could help me.
    Here's the situation.
    You are in jail and have been sentenced to death tomorrow, however there's a way out.
    you're given 12 red balls and 12 blue balls with 2 urns, you have to fully distribute the balls between these two urns. the executioner will then select an urn and draw a ball
    if it is blue you live and if it is red you die.
    suppose you place k balls in to the first urn and 24-k in to the second
    suppose you put i blue balls in to the first urn and 12-i in to the second
    Let A be the event that you live.
    By conditioning on the urn that the executioner chooses, and using the law of total probability show that:

    P(A) = (12i - ki + 6k) / k(24-k)

    please could you explain the steps used so i can try and understand the solution and method.
    Thank you in advance
    Luke
    Start by naming things

    p = the probability that the executioner chooses urn 1, into which you have placed k balls, i of them blue.

    (1 - p) = the probability that the executioner chooses urn 2, into which you have placed 24 - k balls, 12 - i of them blue.

    q = the probability that executioner chooses blue ball given that executioner chooses urn 1 = $\dfrac{i}{k}.$

    r = the probability that executioner chooses blue ball given that executioner chooses urn 2 = $\dfrac{12 - i}{24 - k}.$

    Now, in terms of p, q, and r, what is the probability of

    the executioner picking a blue ball from urn 1,

    the executioner picking a blue ball from urn 2, and

    the executioner picking a blue ball from one urn or the other.

    So what is the hidden assumption to arrive at the answer given?
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  3. #3
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    Re: Conditional Probability and law of total probability.

    The probability of picking a blue ball from urn 1 is : i/k or q
    from urn 2 : r
    then from either one, P(r intersect q) so rq, which gives you (i/k)*(12-i/24-k)
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