# On the number of trials needed to obtain k consecutive successes

*Steve Drekic* and
*Michael Z. Spivey*

*Statistics & Probability Letters*, 2021, vol. 176, issue C

**Abstract:**
A sequence of independent Bernoulli trials, each of which is a success with probability p, is conducted. For k∈Z+, let Xk be the number of trials required to obtain k consecutive successes. Using techniques from elementary probability theory, we present a derivation which ultimately yields an elegant expression for the probability mass function of Xk, and is simpler in comparison to what is found in the literature. Following this, we use our derived formula to obtain explicit closed-form expressions for the complementary cumulative distribution function and the nth factorial moment of Xk.

**Keywords:** Bernoulli trials; Consecutive successes; Factorial moments; Generating function; Polynomial coefficients; Bell polynomials (search for similar items in EconPapers)

**Date:** 2021

**References:** Add references at CitEc

**Citations:** Track citations by RSS feed

**Downloads:** (external link)

http://www.sciencedirect.com/science/article/pii/S0167715221000948

Full text for ScienceDirect subscribers only

**Related works:**

This item may be available elsewhere in EconPapers: Search for items with the same title.

**Export reference:** BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text

**Persistent link:** https://EconPapers.repec.org/RePEc:eee:stapro:v:176:y:2021:i:c:s0167715221000948

**Ordering information:** This journal article can be ordered from

http://www.elsevier.com/wps/find/supportfaq.cws_home/regional

https://shop.elsevie ... _01_ooc_1&version=01

**DOI:** 10.1016/j.spl.2021.109132

Access Statistics for this article

Statistics & Probability Letters is currently edited by *Somnath Datta* and *Hira L. Koul*

More articles in Statistics & Probability Letters from Elsevier

Bibliographic data for series maintained by Catherine Liu ().