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. 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 > > -- > You received this message because you are subscribed to the Google Groups > "sage-support" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-support+unsubscr...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sage-support/fd3261d8-3ee5-48a8-ba6e-40530790b92f%40colby.edu. -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/ZdkZPW_X5wcSJI_O%40hilbert.
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