l'intégration des équations différentielles aux
différences finies, et sur leur usage dans la théorie des
imagine a solid composed
of a number n
of faces perfectly equal, and which I
to have the probability that, by
a number x
of casts, I will bring about these
n faces in
Let be the probability. Laplace obtains the recurrence relation subject to the initial conditions that for and .
Example 1. Laplace solves this in the case of only. We must have and .
When do we have an even wager that the sequence 12 will appear? When .
Example 2. This problem is more difficult with . The initial conditions are that , , and . It is for reason of difficulty, I suspect, that Laplace stopped with .
This solution can be simplified and the use of limits be avoided. Suppose is a root of . Then divides and the factorization of this polynomial is
Thus the other two roots can be expressed in terms of the first as and Moreover, we may rewrite as
We may then substitute this new expression for in the solution given above.
The problem is completed if we then sum over the three roots of , substituting them successively for . Since the polynomial is cubic, we can find the roots exactly. These roots are
This prospect is not fun.
An alternative is to compute the values recursively . We do this in general with the procedure .
> G:=proc(n,x::nonnegint) options remember;
> if x<n then 0 elif x=n then 1/n^n else evalf(G(n,x-1)-G(n,x-3)/n^n+1/n^n); fi; end:
For example , let's consider a die with 6 sides. In how many tosses do we have an even wager that the faces will appear in consecutive order?