2 years ago

Probability and Statistics

Lecture-1

Let us consider a problem involving probability. A communication system is to consist of $n(50)$ seemingly identical antennas that are to be lined up in a linear order.
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The resulting system will then be able to receive all incoming signals and will be called functional as long as no two consecutive antennas are defective . If it turns out that exactly $m(20)$ of $n(50)$ antennas are defective, what is the probability that the resulting system will be functional ?For instance, in the special case where n=4, and m=2 there are 6 possible system configurations, namely, 0 1 1 0, 0 1 0 1,1 0 1 0 , 0 0 1 1, 1 0 0 1, 1 1 0 0
where 1:= non-defective and 0:= defective. First 3 arrangements in our listing functional and remaining 3 are not, so it seems reasonable to take $\frac{3}{6}=\frac{1}{2}$ as the desired probability. In the case of general $n$ and $m$, we could compute the probability that the system is functional in a similar fashion, as long as we could list them. But From for general large m , and n the number of configurations may be so large to list them all. So it would be useful to have an effective method for counting the number of ways that things can occur. In fact many problems in probability theory can be solved simply by counting the number of different ways that a certain event can occur. The mathematical theory of counting is formally known as Combinatorics.

Combinatorics:

Combinatorics is concerned with arrangements of the objects of a set into patterns satisfying specified rules. There are two types of problems that occur in practice.

• Existence of the arrangement: If one wants to arrange the objects of a set that meets given constrains, it may not be obvious whether such an arrangement is possible. If not, then questions is,under what conditions (both necessary and sufficient) such an arrangement is possible.
• Enumeration of arrangements: If an arrangement is possible, there may be several ways of achieving it. If so one might count them effectively.

Problem(a)(Existence of arrangement):A chess master who has 11 weeks to prepare for tournament decides to play at least one game every day but, in order not to tire himself, he decides not to play more than 12 games during any calender week. Show that there exists a consecutive days during which the chess master will have played exactly 21 games.