On Friday, February 23, 2024 at 10:16:37 PM UTC [email protected] wrote:
On Fri, Feb 23, 2024 at 05:00:42PM -0500, Fernando Gouvea wrote:
> In an introductory probability class, one computes the probability of
> getting all of n possible coupons in r individual purchases. The naive
> approach with inclusion-exclusion leads to the awful formula
>
> f(n,r) = \sum_{k=0}^n \binom{n,k} \frac{(n-k)^r}{n^r}
>
> Computing this in GP is straightforward:
>
> (10:32) gp > g(n,k,r)=(-1)^(k)*binomial(n,k)*(n-k)^r/n^r
> %3 = (n,k,r)->(-1)^(k)*binomial(n,k)*(n-k)^r/n^r
> (16:43) gp > f(n,r)=sum(k=0,n,1.0*g(n,k,r))
> %4 = (n,r)->sum(k=0,n,1.0*g(n,k,r))
> (16:43) gp > f(6,12)
> %5 = 0.43781568062117902081322291656082236787
> (16:43) gp > f(50,100)
> %6 = 0.00016616318861823418331310572392871711604
> (16:45) gp > f(50,200)
> %7 = 0.39828703196689437232172533821416865689
> (16:47) gp > f(365,2236)
>
> Doing it in SageMath seems to be much more delicate. This, for example,
> fails:
>
> sage: g(n,k,r)=(-1)^(k)*binomial(n,k)*(n-k)^r/n^r
> sage: f(n,r)=sum(1.0*g(n,k,r),k,0,n)
> sage: f(365,2000).unhold()
> 0.0
>
> It works better if I rewrite the function with ((n-k)/n)^r, of course. I
> presume the issue is that the default precision for real numbers is not
> enough to handle the computation. How do I tell SageMath to use a higher
> precision?
here is how to do it exactly, and convert the result to a float.
sage: sage: g(n,k,r)=(-1)^(k)*binomial(n,k)*(n-k)^r/n^r
....: sage: f(n,r)=sum(g(n,k,r),k,0,n)
....: sage: f(50,200).unhold().n()
0.398287031966894
....: sage: f(365,2000).unhold().n()
0.216119451633218
There should be less brute-force ways too,
bu this is the 1st that comes to mind.
the normal Python way, without any symbolic sum, would be like this:
sage: sage: g(n,k,r)=(-1)^(k)*binomial(n,k)*(n-k)^r/n^r
....: sage: def f(n,r): return math.fsum([1.0*g(n,k,r) for k in range(n+1)])
....: sage: f(365,2000)
0.21611945163321847
HTH
Dima
HTH
Dima
>
> Fernando
>
> --
> =============================================================
> Fernando Q. Gouveahttp://www.colby.edu/~fqgouvea
> Carter Professor of Mathematics
> Dept. of Mathematics
> Colby College
> 5836 Mayflower Hill
> Waterville, ME 04901
>
> Reached by phone, a Westbury woman who identified herself as Onorato's
> granddaughter said he was a retired businessman. He lived alone and
> never married or had children, she said.
> -- New York Newsday, December 27, 2003
>
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