Probability with Recurrence Relation

[chrisyuen]

Homework Statement

Box A contains three white balls and one red ball while box B contains four white balls. One ball is randomly drawn from each box and the two balls are then randomly put back into the boxes so that each box still contains four balls. This process is performed n times. Let P_k be the probability that the red ball is in box A after the process is performed k times, where 1<=k<=n.

(a) Find P₁.
(b) Show that P_k=3/4 P_k-1 + 1/8, where 2<=k<=n. Hence find P_k in terms of k.
(c) Find the limit of the probability that the red ball is finally in box B as n becomes very large.
(d) If the process is performed three times, find the probability that there are exactly two times that the red ball is in box B.

(Answers:
(a) 7/8
(b) (3/8)(3/4)^k-1 + 1/2
(c) 1/2
(d) 57/512)

Homework Equations

Probability Formulae

The Attempt at a Solution

I only knew how to solve part (c) if I set P_k = P_k-1 = P when n --> infinity while

I don't know how can I start the part (a) of this question.

Can anyone tell me how to draw a tree diagram first?

Thank you very much!

 

[Esseintes]

The forum kept deleting my formatting so I put the tree in this picture.

http://sites.google.com/site/blackburnt/_/rsrc/1238089640948/Home/tree.JPG
Does this help?

 

[chrisyuen]

The forum kept deleting my formatting so I put the tree in this picture.

http://sites.google.com/site/blackburnt/_/rsrc/1238089640948/Home/tree.JPG
Does this help?

I got it!

Thank you very much!