Hard probability question (cambridge exam question)

[sweetpotatoes]

original link: http://www.maths.cam.ac.uk/teaching/pastpapers/2001/Part_IA/PaperIA_2.pdf"

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 don't know even how to start, could anyone help?

 

[bpet]

This question is extremely confusing and I don't know even how to start, could anyone help?

To get started on the first part and to help your understanding, try sketching the probability tree for a single gun.

HTH

 

[sweetpotatoes]

To get started on the first part and to help your understanding, try sketching the probability tree for a single gun.

HTH

To get started on the first part and to help your understanding, try sketching the probability tree for a single gun.

HTH

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" referring to? is it N?
How could we use the property that the distribution is preserved

Thank you!

 

[bpet]

... I don't really understand what he means for "Show that if after each move this distribution is preserved, then p = 3/7."

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.