Probability with Recurrence Relation

In summary, the conversation discusses the probability of a red ball being in box A after a random selection process is performed multiple times. The probability is represented by Pk, with P1 being 7/8. Part (b) shows that Pk can be calculated using the formula (3/8)(3/4)k-1 + 1/2 when k is between 2 and n. Part (c) asks for the limit of the probability that the red ball is in box B as n becomes very large, which is found to be 1/2. Finally, part (d) asks for the probability of the red ball being in box B exactly two times out of three, which is calculated to be
  • #1
chrisyuen
56
0

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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  • #2
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:
  • #3
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:

1. What is a recurrence relation?

A recurrence relation is a mathematical equation that describes a sequence of numbers or events, where each term is defined in terms of previous terms. It is often used to model situations in which a quantity changes over time or depends on previous values.

2. How is probability related to recurrence relations?

In probability, a recurrence relation is used to calculate the probability of a particular event occurring multiple times in a sequence. It allows us to determine the probability of a certain outcome based on the probability of previous outcomes. This is especially useful in situations where a random event is repeated multiple times.

3. Can you give an example of a recurrence relation in probability?

One example is the binomial distribution, which is often used to model the number of successes in a series of independent trials. The probability of getting exactly k successes in n trials is given by the recurrence relation P(k) = (n choose k) * p^k * (1-p)^(n-k), where p is the probability of success in each trial.

4. How can we solve a recurrence relation in probability?

There are various methods for solving recurrence relations, such as substitution, iteration, and generating functions. In probability, we often use the principle of inclusion-exclusion to solve recurrence relations involving multiple events or conditions. Additionally, computer programs such as recursion or dynamic programming can be used to solve more complex recurrence relations.

5. What are some real-life applications of probability with recurrence relations?

Probability with recurrence relations has many practical applications in fields such as finance, genetics, and computer science. For example, it can be used to analyze the performance of stock portfolios, predict the likelihood of genetic traits or diseases, and design efficient algorithms for data processing or machine learning. It is also commonly used in game theory to calculate optimal strategies in repeated games.

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