Coin Tossing by Stefan Hollos & J. Richard Hollos

Author:Stefan Hollos & J. Richard Hollos [Hollos, Stefan & Hollos, J. Richard]
Language: eng
Format: epub
Tags: generating functions, probability theory, distributions, stochastic processes, tutorial, mathematics, markov chains
ISBN: 9781887187398
Publisher: Abrazol
Published: 2019-05-01T06:00:00+00:00

Run Occurrence Probability

What is the probability that a run occurs on toss n? The fn values that we have been looking at give the probability that a run occurs for the first time at toss n. Now we simply want the probability that it occurs at toss n, not necessarily for the first time. Let an be the probability that a run of length r appears on the nth toss then we can get a recursion equation for an by looking at all the ways you can have r heads in a row at toss n. If the run occurs at toss n then by definition there are r heads in a row and this has probability an. If the run occurs at toss n − 1 followed by a head then there will also be r heads in a row and this has probability pan − 1. Continuing on like this you can go up to having the run occur at toss n − r + 1 followed by r − 1 heads to have r heads in a row at toss n. Summing up all the possibilities gives the following equation

(164)

This equation is of course only valid for n greater than or equal to r with the initial conditions being an = 0 for n < r. To find the generating function just multiply both sides of the equation by zn and sum from n = r to ∞ (note that a0 is defined as being equal to 1 here). The generating function is then

(165)

The an probabilities can be related to the fn probabilities by looking at all the ways a run can occur on the nth toss. It can occur for the first time on the nth toss with probability fn. It can occur at toss 1 and not reoccur until n − 1 tosses later with probability fn − 1a1 or it can occur at toss 2 and not reoccur until n − 2 tosses later with probability fn − 2a2 and so on. Summing up these probabilities gives

(166)

This equation is valid for n ≥ 1. Multiply both sides by zn and sum from n = 1 to ∞ and you get

(167)

So the two generating functions are related to each other as follows

(168)

(169)

Equation (168) could have been anticipated since when a run occurs at toss n, it is either the first, second, third or higher occurrence, therefore an must be equal to

(170)

This means the generating function for an must be equal to

(171)

which is just the expansion of equation (168)

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.