Absorption Probability Calculator

Calculator inputs

Number of states

Choose between 2 and 10 states.

Starting state

This state begins the chain.

Decimal precision

Higher values show more detailed decimals.

Row tolerance

Used for row-sum and absorbing-state checks.

Input style

Fractions are converted automatically.

Sample shortcut

Useful after changing the state count.

State labels

Label for state 1

Label for state 2

Label for state 3

Label for state 4

Transition matrix

Enter a valid stochastic matrix. Every row must sum to 1. Absorbing states should have 1 on the diagonal and 0 elsewhere.

From \ To	S1	S2	S3	S4
S1
S2
S3
S4

Example data table

This sample uses four states. States S3 and S4 are absorbing. Start from S1 to see how absorption probabilities are distributed.

State	To S1	To S2	To S3	To S4	Type
S1	0.50	0.25	0.25	0.00	Transient
S2	0.30	0.40	0.10	0.20	Transient
S3	0.00	0.00	1.00	0.00	Absorbing
S4	0.00	0.00	0.00	1.00	Absorbing

Formula used

This calculator analyzes an absorbing Markov chain. After arranging states into transient and absorbing groups, the transition matrix is written in canonical form:

P = [ Q R ; 0 I ]

Here, Q contains transient-to-transient probabilities, and R contains transient-to-absorbing probabilities.

The fundamental matrix is:

N = (I - Q)^-1

The absorption probability matrix is:

B = N × R

Each row of B gives the probability of ending in each absorbing state when the chain starts from a particular transient state.

Expected steps before absorption are computed with t = N × 1, where 1 is a column vector of ones.

How to use this calculator

Choose the total number of states.
Label the states if you want custom names.
Select the starting state.
Enter the transition probabilities for every row.
Make sure each row sums to exactly 1.
For absorbing states, place 1 on the diagonal and 0 elsewhere.
Click Calculate Absorption Probabilities.
Review the summary cards, probability table, advanced matrices, and the graph.
Use the CSV or PDF buttons to export the result block.

Frequently asked questions

1. What is an absorbing state?

An absorbing state is a state that, once entered, cannot be left. Its transition row has a 1 on its own diagonal entry and 0 everywhere else.

2. Why must each row sum to 1?

Each row represents all possible next moves from one state. Because one of those moves must occur, the total probability across the row must equal 1.

3. What does the fundamental matrix mean?

The fundamental matrix N measures expected visits to transient states before absorption. It is central for computing both absorption probabilities and expected absorption time.

4. Can the starting state already be absorbing?

Yes. In that case, the process has already finished. The selected absorbing state gets probability 1, every other absorbing state gets probability 0, and expected steps equal 0.

5. Why did the calculator show a validation error?

Validation usually fails when a row does not sum to 1, a probability falls outside 0 to 1, or no absorbing state exists in the matrix.

6. Can I enter fractions instead of decimals?

Yes. Simple fractions such as 1/2, 3/10, and 7/20 are accepted. The calculator converts them into decimals before running the matrix calculations.

7. What if there are multiple absorbing states?

That is a standard use case. The result table lists the probability of eventually ending in each absorbing state from the chosen starting state.

8. Where is this calculator useful?

It is useful in stochastic processes, game states, reliability models, queueing systems, risk studies, genetics, and any process that eventually terminates in final states.