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Unit 18: The Weak Law

18.3 Self Assessment Notes

1. ............... generalized Bernoulli’s theorem around 1800, and in 1866 Tchebychev discovered
the method bearinghis name.

2. In ............... the French mathematician Emile Borel proved adeeper theorem known as the
strong law of large numbers that further generalizes Bernoulli’s theorem.
3. In ............... Kolmogorov derived conditions that were necessary and sufficient for a set of
mutually independent random variables to obey the law of large numbers.
4. A ............... of this result due to Borel and Cantellistates that the above ratio k/n tends to p
not only in probability, but with probability 1. This is the strong law of large numbers
(SLLN).
5. The strong law of large numbers states that if {e } is a sequence of ............... to zero, then
n
  k 
 P   p   n   
n 1  h 


18.4 Review Questions

1. 2n red cards and 2n black cards (all distinct) are shuffled together to form a single deck,
and then split into half. What is the probability that each half will contain n red and n black
cards?
2. 3n red cards and n black cards (all distinct) are shuffled together to form a single deck, and
then split into half. What is the probability that each half will contain n red and n black
cards?
3. 4n red cards and 4n black cards (all distinct) are shuffled together to form a single deck,
and then split into half. What is the probability that each half will contain n red and n black
cards?

4. n red cards and 2n black cards (all distinct) are shuffled together to form a single deck, and
then split into half. What is the probability that each half will contain n red and n black
cards?

Answers: Self Assessment

1. Poisson 2. 1909 3. 1926 4. stronger version
5. positive numbers converging

18.5 Further Readings

Books Sheldon M. Ross, Introduction to Probability Models, Ninth Edition, Elsevier
Inc., 2007.
Jan Pukite, Paul Pukite, Modeling for Reliability Analysis, IEEE Press on
Engineering of Complex Computing Systems, 1998.

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