Skip to main content
SpringerLink
Log in

How many samples does it take to see all the balls in an urn?

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

Let an urn containN balls, numbered from 1 toN. A random number of balls are drawn without replacements from the urn, their numbers are noted and the balls are then returned to the urn. This is done repeatedly, the sample sizes being independent identically distributed. Letv be the number of samples needed to see all the balls. A simple approximation forEv and the asymptotic distribution ofv asN → ∞ are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. A. Markoff,Wahrscheinlichkeitsrechnung, Leipzig-Berlin (1912).

  2. V. F. Kolchin, B. A. Sevast'yanov, and V. P. Chistyakov,Random Allocations, Wiley, New York (1978).

    Google Scholar 

  3. I. N. Kovalenko, A. A. Levitskaya, and M. N. Savchuk,Selected Problems of Probabilistic Combinatorics [in Russian], Naukova Dumka, Kiev (1986).

    Google Scholar 

  4. V. A. Ivanov, G. I. Ivchenko, and Yu. I. Medvedev, “Discrete problems of probability theory,” in:Probability Theory. Mathematical Statistics. Theoretical Cybernetics [in Russian], Vol. 22, Itogi Nauki i Tekhniki, VINITI, Moscow (1984), pp. 3–60.

    Google Scholar 

  5. G. Polya, “Eine Wahrscheinlichkeitsaufgabe in der Kundenwerbung,”Z. Angew. Math. Mech.,10, No. 1–3, 96–97 (1930).

    MATH  Google Scholar 

  6. A. Bekessy, “A lottojatekal kapesolatos nehany cellabetoltesi problemazol. II,”Mat. Lapok,16, 57–67 (1965).

    MathSciNet  Google Scholar 

  7. G. I. Ivchenko and Yu. I. Medvedev, “Asymptotic behavior of the number of particle batches in the classical occupancy problem,”Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.],11, No. 4, 701–708 (1966).

    MathSciNet  Google Scholar 

  8. G. I. Ivchenko, “Some limit theorems in the allocation scheme,” in:Some Applied Problems of Probability Theory and Mathematic Statistics [in Russian], Proc. MIEM, MIEM, Moscow (1973), pp. 111–119.

    Google Scholar 

  9. T. M. Sellke, “How many samples does it take to see all the balls in a box?,”Ann. Appl. Probab.,5, No. 1, 294–309 (1995).

    MATH  MathSciNet  Google Scholar 

  10. W. Feller,An Introduction to Probability Theory and its Applications, Vol. I, Wiley, New York-Chichester-Brisbane (1957).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State Institute of Electronics and Mathematics, USSR

    G. I. Ivchenko

Authors
  1. G. I. Ivchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 58–63, July, 1998.

This research was supported by the Russian Foundation for Basic Research under grant No. 97-01-00387.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ivchenko, G.I. How many samples does it take to see all the balls in an urn?. Math Notes 64, 49–54 (1998). https://doi.org/10.1007/BF02307195

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02307195

Key words

Search

Navigation