original link: http://www.maths.cam.ac.uk/teaching/pastpapers/2001/Part_IA/PaperIA_2.pdf" [Broken](adsbygoogle = window.adsbygoogle || []).push({});

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 X_{j}(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?

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# Hard probability question (cambridge exam question)

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