1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6

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1 Name: Math 4 ctivity 9(Due by EOC Dec. 6) Dear Instructor or Tutor, These problems are designed to let my students show me what they have learned and what they are capable of doing on their own. Please allow them to work the problems on their own! If you would like to help them with similar problems, here are the related homework problems: pg. 449: 7- odd, 9, 4, 4, 49, 5. Thanks! Dear Student, Don t even think about asking for help on the problems in this activity! The recommended homework problems have answers in the book, so do them first for practice!!! Complete the probability distribution and histogram for the following random variables, and determine the expected value.(-). Two fair dice are rolled and the random variable, x, is the sum of the faces showing. Second die First die,,,,4,5,6,,,,4,5,6,,,,4,5,6 4, 4, 4, 4,4 4,5 4,6 5, 5, 5, 5,4 5,5 5,6 6, 6, 6, 6,4 6,5 6,6 x P x

2 . Two names are randomly drawn from a hat without replacement. Three of the names in the hat are ggies, and the other two are Longhorns. Let the random variable, x, be the total number of ggies selected. The sample space for this experiment is L, L, L, L, L, L,,,, L, L x 0 P x 0 0

3 Find the expected winnings for the following games of chance.(-4). Numbers is a game in which you bet $ on any three-digit number from 000 to 999. If your number is randomly selected, you get $500. winnings $ $499 P winnings 4. In Keno, the house has a pot containing 80 balls, each marked with a different number from to 80. You buy a ticket for $ and mark one of the 80 numbers on it. The house then selects 0 numbers at random. If your number is among the 0, you get $.0. winnings $ $.0 P winnings You win the $.0 if your number is one of the 0 numbers selected. The probability that your number is the first number selected is ; the probability that your number is the 80 second number selected is ; and so forth. So the probability that your number is one of the 80 0 numbers selected is times

4 Bonus#: miner is trapped in a mine containing three doors. The first door leads to a tunnel that will take him to safety after hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him to the mine after 7 hours. If we assume that the miner always chooses a door randomly, but won t choose a door more than once, what is the expected length of time until he escapes from the mine? X p X?? 6 First door Second door Third door of hours First door Third door First door Second door of 8 hours of 5 hours of 0 hours of 5 hours Bonus#: fter a terrible battle, it is found that of 00 soldiers, 70 have lost an eye, 75 an ear, 80 an arm, and 85 a leg. t least how many of the 00 soldiers have lost all four?

5 Bonus#: Conditional Expected Value: Sometimes it is easier to calculate E X by using conditional expected values. Suppose the random variable Y takes on the values,, or, and the random variable X takes on the values 4 or 5. Then here are the definitions of the conditional expected values of X: From these and the definition of conditional probability we can conclude that So we get that E X Y P Y E X Y P Y E X Y P Y 4P X 4 Y P X 4 Y P X 4 Y 5P X 5 Y P X 5 Y P X 5 Y 4 P X 4 5 Px 5 E X So the conditional expectation formula is E X EX Y PY EX Y PY EX Y PY. You may use this idea to solve the following problem: miner is trapped in a mine containing three doors. The first door leads to a tunnel that will take him to safety after hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him to the mine after 7 hours. If we assume that the miner is always equally likely to select doors,, and, what is the expected length of time until he escapes from the mine? {Hint: Let X be the number of hours until the miner escapes. door door door door door door E X E X P E X P E X P P E X P E X P E X E X door 5 door 7 door 5 7 }