HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-991102-A

Probability model

a)

edge Let us denote the probability of the transition from condition $i$ to condition $j$ with $p_{i\rightarrow j}$. In the simplest case all of the probabilities are equal, but usually $0 < p_{i\rightarrow j} < 1$.

b)

way Going on subsequent edges the probabilities are multiplied, so the probability is multiplicative.

c)

between two nodes In this case for the resultant probability $p_{i,j}$ we have $p_{i,j} = \sum p_{i\rightarrow j}$, with $p_{i,j} \le 1$. Of course, usually $p_{i,j} \ne p_{j,i}$.

d)

between a node and a set (of nodes) Similarly, as in point c).

EXAMPLE Let us consider a system in the whole-type probability model with conditions $c_1$, $c_2$ and $c_3$ and the transitions

trans. 1:

$c_1 \longrightarrow c_2 \longrightarrow c_3$ $p_{c_1 \rightarrow c_2}=0.4 \quad p_{c_2 \rightarrow c_3}=0.2$

trans. 2:

$c_1 \longrightarrow c_1 \longrightarrow c_3$ $p_{11 \rightarrow 11}=0.5 \quad p_{11 \rightarrow 13}=0.1$

The probability of transition 1 is

$$p_{c_1,c_3}^1=p_{c_1 \rightarrow c_2} \cdot p_{c_2 \rightarrow c_3}= 0.4 \cdot 0.2=0.08.$$

The probability, that from $c_1$ in at most two steps we arrive in $c_3$ is

$$p_{c_1,c_3}^2= p_{c_1 \rightarrow c_3}+ p_{c_1 \rightarrow ... ...row c_1} \cdot p_{c_1 \rightarrow c_3} = 0.1+0.08+0.05= 0.23.$$

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-991102-A