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Chapter 1: Q3E (page 15)
Prove De Morgan鈥檚 laws (Theorem 1.4.9).
\({\left( {A \cup B} \right)^c} = {A^c} \cap {B^c}\) and \({\left( {A \cap B} \right)^c} = {A^c} \cup {B^c}.\)
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Prove that for every two events A and B, the probability that exactly one of the two events will occur is given by the expression
\(\Pr \left( A \right) + \Pr \left( A \right) - 2\Pr \left( {A \cap B} \right)\)
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
If 50 percent of the families in a certain city subscribe to the morning newspaper, 65 percent of the families subscribe to the afternoon newspaper, and 85 percent of the families subscribe to at least one of the two newspapers, what percentage of the families subscribe to both newspapers?
A point (x,y) is to be selected from the squareScontaining all points (x,y) such that 0鈮�x鈮�1 and 0鈮�y鈮�1. Suppose that the probability that the selected point willbelong to each specified subset ofSis equal to the area ofthat subset. Find the probability of each of the followingsubsets:
(a) the subset of points such that\({\left( {x - \frac{1}{2}} \right)^2} + {\left( {y - \frac{1}{2}} \right)^2} \ge \frac{1}{4}\)
(b) the subset of points such that\(\frac{1}{2} < x + y < \frac{3}{2}\)
(c) the subset of points such that\(y \le 1 - {x^2}\)
(d) the subsetof points such thatx=y.
If seven balanced dice are rolled, what is the probability that each of the six different numbers will appear at least once?
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