Page 40 - Probability and Statistical Inference

1. Nations of Probability 17

We use the convention to interpret , whatever be the positive integer n.
The following result uses these combinatorial expressions and it goes by
the name
Theorem 1.4.4 (Binomial Theorem) For any two real numbers a, b and
a positive integer n, one has

Example 1.4.8 Suppose that a fair coin is tossed five times. Now the out-
comes would look like HHTHT, THTTH and so on. By the fundamental rule of
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counting we realize that the sample space S will consist of 2 (=2× 2 × 2 × 2 ×
2), that is thirty two, outcomes each of which has five components and these
are equally likely. How many of these five “dimensional” outcomes would in-
clude two heads? Imagine five distinct positions in a row and each position will
be filled by the letter H or T. Out of these five positions, choose two positions
and fill them both with the letter H while the remaining three positions are
filled with the letter T. This can be done in ways, that is in (5)(4)/2 = 10
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ways. In other words we have P(Two Heads) = / 2 = 10/32 = 5/16. !
Example 1.4.9 There are ten students in a class. In how many ways can the
teacher form a committee of four students? In this selection process, naturally
the order of selection is not pertinent. A committee of four can be chosen in
ways, that is in (10)(9)(8)(7)/4! = 210 ways. The sample space S would then
consist of 210 equally likely outcomes. !
Example 1.4.10 (Example 1.4.9 Continued) Suppose that there are six men
and four women in the small class. Then what is the probability that a ran-
domly selected committee of four students would consist of two men and two
women? Two men and two women can be chosen in ways, that is in 90
ways. In other words, P(Two men and two women are selected) = / =
90/120 = 3/7. !
Example 1.4.11 John, Sue, Rob, Dan and Molly have gone to see a
movie. Inside the theatre, they picked a row where there were exactly five
empty seats next to each other. These five friends can sit in those chairs in
exactly 5! ways, that is in 120 ways. Here, seating arrangement is naturally
pertinent. The sample space S consists of 120 equally likely outcomes. But
the usher does not know in advance that John and Molly must sit next to
each other. If the usher lets the five friends take those seats randomly, what
is the probability that John and Molly would sit next to each other? John
and Molly may occupy the first two seats and other three friends may
permute in 3! ways in the remaining chairs. But then John and Molly may