Probability with Recurrence Relation

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SUMMARY

The discussion centers on calculating probabilities related to a red ball in two boxes through a recurrence relation. The initial probability P1 of the red ball being in box A after one draw is established as 7/8. The recurrence relation is defined as Pk = (3/4)Pk-1 + 1/8 for k ≥ 2, leading to the general formula Pk = (3/8)(3/4)^(k-1) + 1/2. As n approaches infinity, the probability that the red ball is in box B converges to 1/2. Additionally, the probability of the red ball being in box B exactly two times after three draws is calculated as 57/512.

PREREQUISITES
  • Understanding of basic probability concepts
  • Familiarity with recurrence relations
  • Knowledge of tree diagrams for probability analysis
  • Ability to manipulate algebraic expressions
NEXT STEPS
  • Study recurrence relations in probability theory
  • Learn how to construct and interpret tree diagrams for probability problems
  • Explore the concept of limits in probability as n approaches infinity
  • Investigate binomial probability distributions for similar problems
USEFUL FOR

This discussion is beneficial for students studying probability theory, educators teaching recurrence relations, and anyone interested in advanced probability calculations involving multiple events.

chrisyuen
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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 Pk 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 Pk=3/4 Pk-1 + 1/8, where 2<=k<=n. Hence find Pk 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 Pk = Pk-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!
 
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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?
 
Last edited by a moderator:
Esseintes said:
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!
 
Last edited by a moderator:

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