91Ó°ÊÓ

Three dice are tossed, one red, one blue, and one green. What outcomes make up the event \(A\) that the sum of the three faces showing equals 5 ?

A poker deck consists of fifty-two cards, representing thirteen denominations ( 2 through ace) and four suits (diamonds, hearts, clubs, and spades). A five- card hand is called a flush if all five cards are in the same suit but not all five denominations are consecutive. Pictured below is a flush in hearts. Let \(N\) be the set of five cards in hearts that are not flushes. How many outcomes are in N? [Note: In poker, the denominations \((\mathrm{A}, 2,3,4,5)\) are considered to be consecutive (in addition to sequences such as \((8,9\), \(10, \mathrm{~J}, \mathrm{Q}))\).]

Let \(P\) be the set of right triangles with a \(5^{\prime \prime}\) hypotenuse and whose height and length are \(a\) and \(b\), respectively. Characterize the outcomes in \(P\).

A woman has her purse snatched by two teenagers. She is subsequently shown a police lineup consisting of five suspects, including the two perpetrators. What is the sample space associated with the experiment "Woman picks two suspects out of lineup"? Which outcomes are in the event \(A\) : She makes at least one incorrect identification?

Consider the experiment of choosing coefficients for the quadratic equation \(a x^{2}+b x+c=0\). Characterize the values of \(a, b\), and \(c\) associated with the event \(A\) : Equation has complex roots.

A probability-minded despot offers a convicted murderer a final chance to gain his release. The prisoner is given twenty chips, ten white and ten black. All twenty are to be placed into two urns, according to any allocation scheme the prisoner wishes, with the one proviso being that each urn contain at least one chip. The executioner will then pick one of the two urns at random and from that urn, one chip at random. If the chip selected is white, the prisoner will be set free; if it is black, he "buys the farm." Characterize the sample space describing the prisoner's possible allocation options (Intuitively, which allocation affords the prisoner the greatest chance of survival?)

Suppose that ten chips, numbered 1 through 10 , are put into an urn at one minute to midnight, and chip number 1 is quickly removed. At one-half minute to midnight, chips numbered 11 through 20 are added to the urn, and chip number 2 is quickly removed. Then at one-fourth minute to midnight, chips numbered 21 to 30 are added to the urn, and chip number 3 is quickly removed. If that procedure for adding chips to the urn continues, how many chips will be in the urn at midnight (157)?

Let \(A\) be the set of five-card hands dealt from a fifty-two-card poker deck, where the denominations of the five cards are all consecutive - for example, (7 of hearts, 8 of spades, 9 of spades, 10 of hearts, jack of diamonds). Let \(B\) be the set of five-card hands where the suits of the five cards are all the same. How many outcomes are in the event \(A \cap B\) ?

Let \(A_{1}, A_{2}, \ldots, A_{k}\) be any set of events defined on a sample space \(S\). What outcomes belong to the event $$ \left(A_{1} \cup A_{2} \cup \cdots \cup A_{k}\right) \cup\left(A_{1}^{C} \cap A_{2}^{C} \cap \cdots \cap A_{k}^{C}\right) $$

Let \(A, B\), and \(C\) be any three events defined on a sample space \(S\). Show that the operations of union and intersection are associative by proving that (a) \(A \cup(B \cup C)=(A \cup B) \cup C=A \cup B \cup C\) (b) \(A \cap(B \cap C)=(A \cap B) \cap C=A \cap B \cap C\)