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Math Help - Selecting balls numbered 1 through 4 from an urn

  1. #1
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    Selecting balls numbered 1 through 4 from an urn

    An urn contains four balls numbered 1 through 4. The balls are selected one at a time without replacement. A match occurs if ball numbered m is the mth ball selected. Let the event Ai denote a match on the ith draw, i = 1,2,3,4.

    a) Show that P(Ai) = 3!/4!

    b) Show that P(Ai Intersection Aj) = 2!/4!

    c) Show that P(Ai Intersection Aj intersection Ak) = 1!/4!

    d) Show that the probability of at least one match is P (A1 U A2 U A3 U A4) = 1 - 1/2! + 1/3! - 1/4!

    e) Extend this exercise so that there are n balls in the urn. Show that the probability of at least one match is:

    P(A1 U A2 U...U An) = 1 - 1/2! + 1/3! - 1/4! +...+ (-1)n+1 = 1 - (1- 1/1! + 1/2! - 1/3! +...+ (-1)n ).
    n! n!

    This is exercise 1.4-19 from Probability and Statistical Inference, Hogg and Tanis 8th edition. If anyone could help me with this problem I would greatly appreciate it.
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  2. #2
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    Re: Selecting balls numbered 1 through 4 from an urn

    You are supposed to show some work.

    Quote Originally Posted by polcareb192 View Post
    a) Show that P(Ai) = 3!/4!
    Of all permutations of 4 elements, you select those that leave i in place. The other 3 elements can come in an arbitrary order.

    Quote Originally Posted by polcareb192 View Post
    b) Show that P(Ai Intersection Aj) = 2!/4!
    Provided that i ≠ j. This is similar to a).

    Quote Originally Posted by polcareb192 View Post
    c) Show that P(Ai Intersection Aj intersection Ak) = 1!/4!
    Provided i, j and k are pairwise distinct.

    Quote Originally Posted by polcareb192 View Post
    d) Show that the probability of at least one match is P (A1 U A2 U A3 U A4) = 1 - 1/2! + 1/3! - 1/4!
    Use the inclusion-exclusion principle.
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  3. #3
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    Re: Selecting balls numbered 1 through 4 from an urn

    ***The underlined terms should be over n!***
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  4. #4
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    Re: Selecting balls numbered 1 through 4 from an urn

    Quote Originally Posted by polcareb192 View Post
    An urn contains four balls numbered 1 through 4. The balls are selected one at a time without replacement. A match occurs if ball numbered m is the mth ball selected. Let the event Ai denote a match on the ith draw, i = 1,2,3,4.

    a) Show that P(Ai) = 3!/4!
    Why in the world write it in such a strange way? 3!/4!= 1/4. And that's easy to show.

    A1 is the event of getting ball number 1 on the first draw: there are 4 balls in the urn so the probability of drawing any specific ball on the first draw is 1/4.

    A2 is the event of getting ball number :2" on the second draw. Since this is "sampling without replacement" in order to get ball "2" on the second draw we must NOT have got it on the first draw. There were 4 balls, 3 of which were not labled "2" so the probability of getting anything except "2" on the first draw is 3/4. Once we have done that, there are 3 balls left, one of which is labeled "2". The probability of getting that ball on the second draw is 1/3 so the probability of getting "ball number "2" on the second draw" is (3/4)(1/3)= 1/4.

    Similarly, to get ball "3" on the third draw we must NOT get it on the first draw, which has probability 3/4, then not get it on the second draw, which has probability 2/3, then get it on the third draw, which has probability 1/2. The probability of getting "ball "3" on the third draw" is (3/4)(2/3)(1/2)= 1/4.

    Of course, the probability of getting the ball marked "4" on the fourth draw is (3/4)(2/3)(1/2)(1/1)= 1/4.

    b) Show that P(Ai Intersection Aj) = 2!/4!
    2!/4!= 1/12 and can be argued in a similar manner to (a).

    c) Show that P(Ai Intersection Aj intersection Ak) = 1!/4!

    d) Show that the probability of at least one match is P (A1 U A2 U A3 U A4) = 1 - 1/2! + 1/3! - 1/4!

    e) Extend this exercise so that there are n balls in the urn. Show that the probability of at least one match is:

    P(A1 U A2 U...U An) = 1 - 1/2! + 1/3! - 1/4! +...+ (-1)n+1 = 1 - (1- 1/1! + 1/2! - 1/3! +...+ (-1)n ).
    n! n!

    This is exercise 1.4-19 from Probability and Statistical Inference, Hogg and Tanis 8th edition. If anyone could help me with this problem I would greatly appreciate it.
    Last edited by HallsofIvy; June 20th 2012 at 08:11 AM.
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    Re: Selecting balls numbered 1 through 4 from an urn

    Quote Originally Posted by HallsofIvy View Post
    Why in the world write it in such a strange way? 3!/4!= 1/4. And that's easy to show.
    I think that the problem author computes the probabilities of events by definition: the cardinality of an event divided by the cardinality of the sample space. (It is assumed that all singletons in the sample space have equal probability.) The sample space here consists of permutations of (1, 2, 3, 4). The event Ai consists of all permutations where the ith element is i, and other elements are arbitrary. Thus, |Ai| = 3!. My guess is that 3! / 4! gives a hint, which is described above, how to compute the answer 1 / 4. This method is arguable easier: it finds P(Ai) for all i at once and it does not use conditional probability.
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