2 red marbles and 8 blue marbles are placed into a bag

1. 1.     2 red marbles and 8 blue marbles are placed into a bag.  Bob mixes up the bag and randomly selects a marble.  He continues to do so, replacing the marble after each selection, until a red marble is selected.                               (12 points)
   1. a.     What is the probability that the first time that a red marble is pulled is on Bob’s 8^th try?
   2. b.     On average, how many marbles will Bob have to pull in order to get a red marble?             (Hint: use math expectation)
2. 2.     Let a fair die be rolled n times. Let’s make an assumption that all rolls of a die are independent. If X and Y are the outcomes of any two of n trials, what is the probability distribution of X+Y?  That is, create a table that contains each unique possible value of X+Y (each value only listed once) and each possibility’s corresponding probability.                                                                       (15 points)
3. 3.     We have a fair ten-sideddie.                                                             (15 points)
   1. a.     Find the math expectation of a single roll. 
   2. b.     Find the math expectation of the numerical sum of 5 rolls.
   3. c.      Find the math expectation of the numerical product (i.e., multiplication) of 7 rolls.
4. are identically distributed independent random variables such that
    and  . What is the standard deviation of their average?  In other words, what is the standard deviation of ?                                                                                                                    (8 points)

1. If the cumulative distribution function of is  given by the function below, then find P (X ≥ 0.75).                                                                                           (10 points)

, if x ≤ 0 
 x², if 0 <  x ≤ 1 
, if  x > 1

1. At the town fair, you can pay $5 to toss a ring at a set of bottles.  If you get a “ringer” on the small mouth bottle, you win $50.  If you get a “ringer” on the medium bottle, you win $10.  If you get a “ringer” on the large bottle, you get your $5 fee back (that is, you break even).  If you miss, you are out the $5 you paid to play.  Ryan is a good shot and his probability of getting a ringer on the small, medium, and large bottles is 10%, 10%, and 20%, respectively.  The probability distribution of Ryan’s winnings in a single game is given below. 

1. Find the math expectation of Ryan’s winningsfor a single game.
2. Find the math expectation of Ryan’s winnings after 3 games.
3. Find the varianceof Ryan’s winnings for a single  game.
4. Find the standard deviation of Ryan’s winnings for a single game.
5. Does it pay for Ryan to play this game at the fair?  Explain.
6. Find the cumulative distribution function of Ryan’s winnings for a single game and draw its graph.                                          

(5 points each for parts a-e and 20 points for part f)