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Hard probability question (cambridge exam question)

  1. Dec 3, 2009 #1
    original link: http://www.maths.cam.ac.uk/teaching/pastpapers/2001/Part_IA/PaperIA_2.pdf" [Broken]

    Question 11F

    Dipkomsky, a desperado in the wild West, is surrounded by an enemy gang and
    fighting tooth and nail for his survival. He has m guns, m > 1, pointing in different
    directions and tries to use them in succession to give an impression that there are several
    defenders. When he turns to a subsequent gun and discovers that the gun is loaded
    he fires it with probability 1/2 and moves to the next one. Otherwise, i.e. when the
    gun is unloaded, he loads it with probability 3/4 or simply moves to the next gun with
    complementary probability 1/4. If he decides to load the gun he then fires it or not with
    probability 1/2 and after that moves to the next gun anyway.
    Initially, each gun had been loaded independently with probability p. Show that if
    after each move this distribution is preserved, then p = 3/7. Calculate the expected value
    EN and variance Var N of the number N of loaded guns under this distribution.

    Hint: it may be helpful to represent N as a sum Xj (1 to m) of random variables
    taking values 0 and 1.

    This question is extremely confusing and I dunno even how to start, could anyone help?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Dec 4, 2009 #2
    To get started on the first part and to help your understanding, try sketching the probability tree for a single gun.

    HTH
     
  4. Dec 5, 2009 #3
    Thanks for the hints.
    For m=1, then initially we have prob of p having it loaded, before the first round we have two case
    loaded: prob of p
    not loaded: 1-p
    after 1st round
    loaded:
    not loaded: 1-p*1/4

    Any idea what should I do next?

    Also, I don't really understand what he means for "Show that if
    after each move this distribution is preserved, then p = 3/7."
    What exactly is the "distribution" refering to? is it N?
    How could we use the property that the distribution is preserved

    Thank you!
     
  5. Dec 6, 2009 #4
    Say the probability after the move is p1, then p1=p. In the first part you'll have derived an expression for p1 in terms of p, and if your algebra is correct then p=3/7 is the only solution to the equation p1=p. To get the probabilities right though, you might need to practice on some simpler probability tree questions first.
     
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