MathDB

Consider a coin with a head toss probability

p

where

0 <p <1

is fixed. Toss the coin several times, the tosses should be independent of each other. Denote by

A_i

the event that of the

i

-th,

(i + 1)

-th,

\ldots

, the

(i+m-1)

-th throws, exactly

T

is the tail. For

T = 1

, calculate the conditional probability

\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)

, and for

T = 2

, prove that

\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)

has approximation in the form

a+ \tfrac{b}{m} + \mathcal{O}(p^m)

m \to \infty

Problem Statement