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Chapter 1: Q9E (page 41)
If n people are seated in a random manner in a row containing 2n seats, what is the probability that no two people will occupy adjacent seats?
The probability that no two people will occupy adjacent seats is \(\frac{{n{!^2} \times \left( {n + 1} \right)}}{{2n!}}\)
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Suppose that the sample space S of some experiment is finite. Show that the collection of all subsets of S satisfies the three conditions required to be called the collection of events.
Consider a state lottery game in which each winning combination and each ticket consists of one set of k numbers chosen from the numbers 1 to n without replacement. We shall compute the probability that the winning combination contains at least one pair of consecutive numbers.
a. Prove that if\({\bf{n < 2k - 1}}\), then every winning combination has at least one pair of consecutive numbers. For the rest of the problem, assume that\({\bf{n}} \le {\bf{2k - 1}}\).
b. Let\({{\bf{i}}_{\bf{1}}}{\bf{ < }}...{\bf{ < }}{{\bf{i}}_{\bf{k}}}\)be an arbitrary possible winning combination arranged in order from smallest to largest. For\({\bf{s = 1,}}...{\bf{,k}}\), let\({{\bf{j}}_{\bf{s}}}{\bf{ = }}{{\bf{i}}_{\bf{s}}}{\bf{ - }}\left( {{\bf{s - 1}}} \right)\). That is,
\(\begin{array}{c}{{\bf{j}}_{\bf{1}}}{\bf{ = }}{{\bf{i}}_{\bf{1}}}\\{{\bf{j}}_{\bf{2}}}{\bf{ = }}{{\bf{i}}_{\bf{2}}}{\bf{ - 1}}\\{\bf{.}}\\{\bf{.}}\\{\bf{.}}\\{{\bf{j}}_{\bf{k}}}{\bf{ = }}{{\bf{i}}_{\bf{k}}}{\bf{ - }}\left( {{\bf{k - 1}}} \right)\end{array}\)
Prove that\(\left( {{{\bf{i}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{i}}_{\bf{k}}}} \right)\)contains at least one pair of consecutive numbers if and only if\(\left( {{{\bf{j}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{j}}_{\bf{k}}}} \right)\)contains repeated numbers.
c. Prove that\({\bf{1}} \le {{\bf{j}}_{\bf{1}}} \le ... \le {{\bf{j}}_{\bf{k}}} \le {\bf{n - k + 1}}\)and that the number of\(\left( {{{\bf{j}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{j}}_{\bf{k}}}} \right)\)sets with no repeats is\(\left( {\begin{array}{*{20}{c}}{{\bf{n - k + 1}}}\\{\bf{k}}\end{array}} \right)\)
d. Find the probability that there is no pair of consecutive numbers in the winning combination.
e. Find the probability of at least one pair of consecutive numbers in the winning combination
For each integer n, let \({{\bf{X}}_{\bf{n}}}\) be a nonnegative random variable with finite mean \({{\bf{\mu }}_{\bf{n}}}\). Prove that if\(\mathop {\lim }\limits_{n \to \infty } {{\bf{\mu }}_{\bf{n}}}{\bf{ = 0}}\), then.
Suppose thatnbalanced dice are rolled. Determine the probability that the numberjwill appear exactlynjtimes(j=1, . . . ,6), wheren1+n2+. . .+n6=n.
A box contains 100 balls, of which rare red. Suppose that the balls are drawn from the box one at a time, at random,without replacement. Determine (a) the probability that the first ball drawn will be red; (b) the probability that the 50th ball drawn will be red, and (c) the probability that the last ball drawn will be red.
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