Juwu u.-....._-_._.. __. (1) Two fair 6-sided dice are rolled. What is the probability (a) that \"doubles\" are rolled (i.e. that the two dice show the same number)? (b) that the sum rolled is 9? (c) that exactly one of the dice shows a \"3\"? (d) that at least one of the dice shows a \"3\"? (2) A standard deck of 52 cards has 13 ranks (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king) and 4 suits (spades, hearts, diamonds, and clubs), such that there is exactly one card for any given rank and suit. The deck is randomly arranged. What is the probability that (a) the top card is an ace or a king. (b) the top card is spades and the second card is clubs. (c) the top card is spades and the second card is an ace. ((1) the top 3 cards are all spades. (e) the top 4 cards include 3 different ranks, with one rank apprears twice (for example, an ace of hearts, a 3 of clubs, a 3 of hearts, and a 7 of spades). (3) A senate committee consists of 5 Republicans, 6 Democrats, and 2 Independents. A subcommittee of 3 members is randomly chosen. What is the probability that the subcommittee (a) consists of 3 Republicans? / (b) consists of 1 Republican, 1 Democrats, and 1 Independent? (4) The letters of the word \"sixteen\" are randomly arranged. What is the probability that the two e's are not next to each other? (5) At a hospital's emergency room, patients are classied and 20% of them are critical, 30% are serious, and 50% are stable. Of the critical ones, 30% die; of the serious, 10% die; and of the stable, 1% die. Given that a patient dies, what is the conditional probability that the patient was classied as critical? (6) Each of the 12 students in a class was given a fair 12sided die. In addition, each student is numbered from 1 to 12. (a) If the students roll their dice, what is the probability that there is at least a \"match\" (e.g. student 4 rolls a 4)? (b) If you are a member of this class, what is the probability that at least one of the other 11 students rolls the same number as you do