# Markov Chains and absorption probabilites

• macca1994
In summary, a single-celled organism contains N particles, some of which are of type A, the others of type B. Daughter cells are formed by cell division, but first each particle replicates itself. The daughter cell inherits N particles chosen at random from the 2i particles of type A and 2N-2i of type B in the parent cell. Find the absorption probabilities and expected times to absorption for the case N = 3.
macca1994

A single-celled organism contains N particles, some of which are of type A, the others of
type B . The cell is said to be in state i where 0<=i<=N if it contains exactly i particles
of type A. Daughter cells are formed by cell division, but rst each particle replicates itself;
the daughter cell inherits N particles chosen at random from the 2i particles of type A
and 2N-2i of type B in the parent cell.

Find the absorption probabilities and expected times to absorption for the case N = 3.

I so far have that the absorbing states are i=0, i=3 but have no idea where to go from there

For N=3, you can calculate the transition matrix manually. Many entries are 0, and some others follow from symmetry, so you just need 2 interesting entries.

how do i calculate the entries though, that's where I'm stuck at the moment, i know of course the lines for starting in state 0 and 3, but have no clue about 1 or 2, once i know that the rest of the question becomes fairly trivial, could you push me in the right direction?

i=1 leads to AABBBB in the cell before splitting. If you randomly pick 3 of them, what is the probability of getting 0 (,1,2,3) times A?

oh is that standard binomial? so probability of going from state 1 to 0 would be (2/3)^3 which is 8/27 then do the same for the other states? or am i missing something?

i really don't understand the probabilities of getting to the other states, do i not need to also consider what the other cell will contain or is that irrelevant?

I think i finally get it, so probability of 0 A's is equal to
(2/3)*(3/5)*(1/2) which is the probability of selecting a B each time
Then follow the same method for 1 A taking into account whether you chose the A first, second or third? I hope that's right

That is correct.

Thanks for the help

I also have difficulty in this question.I have calculated that the probabilty of getting 0 'a's is 1/5...probability of getting 1 'a' is 3/5 and probability of getting 2 'a' is 1/5. What is the transition matrix and what are the absorption probabilities?
thank you

macca1994 said:
I think i finally get it, so probability of 0 A's is equal to
(2/3)*(3/5)*(1/2) which is the probability of selecting a B each time
Then follow the same method for 1 A taking into account whether you chose the A first, second or third? I hope that's right

If you start with AABBBB and pick three at random, you are looking at the "hypergeometric distribution", which is the probability for choosing k of type A from a population of 2A and 4B, when you choose three altogether. See, eg., http://en.wikipedia.org/wiki/Hypergeometric_distribution or http://mathworld.wolfram.com/HypergeometricDistribution.html .

thank you..i have already found the probabilities but how to find the transition matrix please?

Viper7593 said:
thank you..i have already found the probabilities but how to find the transition matrix please?

Work it out for yourself. If you start in state i = 1, what are the possible next states you can reach in one step? What are the probabilities of going to those various states in one step? Once you have answered those questions you will have worked out what is the i = 1 row of the transition matrix. You get the other rows in a similar way. There are no shortcuts; you have to sit down and do it all, step-by-step; and you will only learn how by doing it yourself.

## 1. What is a Markov Chain?

A Markov Chain is a mathematical model that describes a sequence of events where the probability of transitioning from one state to another only depends on the current state and not on any previous states.

## 2. What are the applications of Markov Chains?

Markov Chains have various applications in fields such as economics, finance, biology, and physics. They are used to model and analyze processes that involve random and sequential events.

## 3. What is an absorption probability in a Markov Chain?

An absorption probability in a Markov Chain is the probability of eventually transitioning to a particular state, known as the absorbing state, and staying in that state forever. These probabilities can be calculated using the fundamental matrix of the Markov Chain.

## 4. How is the steady-state distribution of a Markov Chain related to absorption probabilities?

The steady-state distribution of a Markov Chain represents the long-term proportion of time spent in each state. The absorption probabilities can be derived from the steady-state distribution by taking the proportion of time spent in each absorbing state.

## 5. How can Markov Chains be used to predict future events?

By analyzing the transition probabilities and absorption probabilities of a Markov Chain, future events can be predicted. This is because the model assumes that the future state only depends on the current state, making it a useful tool for forecasting.

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