Selection system - Gambling system

[woundedtiger4]

Hi all,

Currently I am trying to understand the concept of Selection System given in Billingsley's book "Probability and Measure". I have attached a pdf file with this post which consists on the scanned pages of this topic. I have understood it completely except one thing that how has the text calculated the values of "N_n". In the attached file there is an example 7.4 in which the values of N_1, N_2, N_3, and N_4 are given but I don't understand that how to calculate them or where have the come from though I do understand there role in the calculation of Y_n.

Thanks in advance.

 

[micromass]

The random variable N_n indicates at what time the n'th bet will take place.

For example, if B₁=0, B₂=0, B₃=1, B₄=0, B₅=1.

Then the first bet will occur in the 3th stage (because 3 is the first number n such that B_n=1). Thus N₁=3.
The second bet will occur in the 5th stage (because 5 is the second number n such that B_n=1). Thus N₂=5.

 

[woundedtiger4]

The random variable N_n indicates at what time the n'th bet will take place.

For example, if B₁=0, B₂=0, B₃=1, B₄=0, B₅=1.

Then the first bet will occur in the 3th stage (because 3 is the first number n such that B_n=1). Thus N₁=3.
The second bet will occur in the 5th stage (because 5 is the second number n such that B_n=1). Thus N₂=5.

thank you sir.