l'intégration des équations différentielles aux
différences finies, et sur leur usage dans la théorie des
suppose two players
A and B, with an equal number m
of écus, playing to this condition,
one who loses will give an écu to the other; let the probability
A winning a trial be p
; let that of B be
q ; but let it
to happen that any of them not win, and let the probability of this be
we ask the probability that the game will end before or at the number
Consider the probability that player A has a stake k at time x . We may compute this through a Markov chain with two absorbing states, entered when the player either loses his stake or gains the entire stake of the other.
Example. Suppose . The transition matrix P is clearly the one having r on the main diagonal excepting the corners, q on the sub-diagonal and p on the super-diagonal. The initial state a is the vector having 1 in the st position.
Warning, the protected names norm and trace have been redefined and unprotected
The probability that player A holds a stake k at time x is given by the value in the k th position of the expression .
Thus, for example, we can examine the game after 4 matches.
To determine the probability that the game end at or before time x , it suffices to sum the first and last components. To this end we define the column vector b with 1 in its first and last position and zeros elsewhere.
Now, is this desired probability.
The game ends at or before time 4 with probability
Suppose now that p , q and r each equal 1/3.