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Chapter 1: Q5E (page 16)

For every collection of events\({A_i}\left( {i \in I} \right)\), show that

\({\left( {\bigcup\limits_{i \in I} {{A_i}} } \right)^c} = \bigcap\limits_{i \in I} {{A_i}^c} \)and\({\left( {\bigcap\limits_{i \in I} {{A_i}} } \right)^c} = \bigcup\limits_{i \in I} {{A_i}^c} \).

Short Answer

Expert verified

We have to prove that \({\left( {\bigcup\limits_{i \in I} {{A_i}} } \right)^c} = \bigcap\limits_{i \in I} {{A_i}^c} \)and \({\left( {\bigcap\limits_{i \in I} {{A_i}} } \right)^c} = \bigcup\limits_{i \in I} {{A_i}^c} \).

Step by step solution

01

Given information

\({A_i}\)is a collection of events, where\(i \in I\).

02

Prove\({\left( {\bigcup\limits_{i \in I} {{A_i}} } \right)^c} = \bigcap\limits_{i \in I} {{A_i}^c} \)

Let x be an arbitrary event, such that,

\(\begin{aligned}{l}x \in {\left( {\bigcup\limits_{i \in I} {{A_i}} } \right)^c}\\ \Rightarrow x \notin {A_i}\left( {for\;any\;i \in I} \right)\\ \Rightarrow x \in A_i^c\;\left( {\forall i \in I} \right)\\ \Rightarrow x \in A_1^c\;{\rm{and}}\;x \in A_2^c\;{\rm{and}}\;x \in A_3^c\;{\rm{and}}\;{\rm{so}}\;{\rm{on}}\\ \Rightarrow {\rm{x}} \in \bigcap\limits_{i \in I} {A_i^c} \end{aligned}\)

Hence, \({\left( {\bigcup\limits_{i \in I} {{A_i}} } \right)^c} = \bigcap\limits_{i \in I} {{A_i}^c} \).

03

Prove\({\left( {\bigcap\limits_{i \in I} {{A_i}} } \right)^c} = \bigcup\limits_{i \in I} {{A_i}^c} \) 

Let y be an arbitrary event, such that,

\(\begin{aligned}{}y \in {\left( {\bigcap\limits_{i \in I} {{A_i}} } \right)^c}\\ \Rightarrow y \notin \bigcap\limits_{i \in I} {{A_i}} \\ \Rightarrow y \in {A_i}^c\;\left( {{\rm{for some }}i \in I} \right)\\ \Rightarrow y \in {A_1}^c{\rm{ or }}y \in {A_2}^c{\rm{ or }}y \in {A_3}^c{\rm{ and so on}}\\ \Rightarrow y \in \bigcup\limits_{i \in I} {{A_i}^c} \end{aligned}\)

Hence, \({\left( {\bigcap\limits_{i \in I} {{A_i}} } \right)^c} = \bigcup\limits_{i \in I} {{A_i}^c} \).

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Most popular questions from this chapter

Let S be the sample space for some experiment. Show that the collection of subsets consisting solely of S and \(\phi \) satisfies the three conditions required in order to be called the collection of events. Explain why this collection would not be very interesting in most real problems.

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.

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?

A deck of 52 cards contains four aces. If the cards are shuffled and distributed in a random manner to four players so that each player receives 13 cards, what is the probability that all four aces will be received by the same

player?

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
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