91Ó°ÊÓ

For \(A=\\{1,2,3,4,5,6,7\\}\), determine the number of a) subsets of \(A\) b) nonempty subsets of \(A\) c) proper subsets of \(A\) d) nonempty proper subsets of \(A\) e) subsets of \(A\) containing three elements f) subsets of \(A\) containing 1,2 g) subsets of \(A\) containing five elements, including 1,2 h) subsets of \(A\) with an even number of elements i) subsets of \(A\) with an odd number of elements

a) If a set \(A\) has 63 proper subsets, what is \(|A| ?\) b) If a set \(B\) has 64 subsets of odd cardinality, what is \(|B| ?\) c) Generalize the result of part (b)

How many arrangements of the letters in CHEMIST have H before \(\mathrm{E}\), or E before \(\mathrm{T}\), or T before M? (Here "before" means anywhere before, not just immediately before.)

A carnival game invites a player to select one card from a standard deck of 52 cards. If the card is a seven or a jack the player is given five dollars. For a king or an ace the player is given eight dollars. The other 36 cards result in the player losing. How much should one be willing to pay to play this game so that it is fair - that is, so that the expected value of the player's net winnings is \(0 ?\)

Suppose that a random variable \(X\) has mean \(E(X)=17\) and variance \(\operatorname{Var}(X)=9\), but its probability distribution is unknown. Use Chebyshev's Inequality to estimate a lower bound for (a) \(\operatorname{Pr}(11 \leq X \leq 23)\); (b) \(\operatorname{Pr}(10 \leq X \leq 24)\); and (c) \(\operatorname{Pr}(8 \leq X \leq 26)\).

The probability that a certain mechanical component fails when first used is \(0.05\). If the component does not fail immediately, the probability it will function correctly for at least one year is \(0.98\). What is the probability that a new component functions correctly for at least one year?

Prove or disprove each of the following for sets \(A, B \subseteq U\). a) \(\mathscr{P}(A \cup B)=\mathscr{P}(A) \cup \mathscr{P}(B)\) b) \(\mathscr{P}(A \cap B)=\mathscr{P}(A) \cap \mathscr{P}(B)\)

Fred rolls a fair die 20 times. If \(X\) is the random variable that counts the number of 6 's that come up during the 20 rolls, determine \(E(X)\) and \(\operatorname{Var}(X)\).

a) How many subsets of \(\\{1,2,3, \ldots, 11\\}\) contain at least one even integer? b) How many subsets of \(\\{1,2,3, \ldots, 12\\}\) contain at least one even integer? c) Generalize the results of parts (a) and (b).

The nine members of a coed intramural volleyball team are to be randomly selected from nine college men and ten college women. To be classified as coed the team must include at least one player of each gender. What is the probability the selected team includes more women than men?