1. 10 red marbles and 10 blue marbles are placed into a bag.

1.       10 red marbles and 10 blue marbles are placed into a bag.  Alex 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.                            (10 points)

a.      What is the probability that the first time that a red marble is pulled is on Alex’s 6^th try?

b.      On average, how many marbles will Alex have to pull in order to get a red marble?           (Hint: use math expectation)

2.      Let a fair die be rolled 2 times. Let’s assume that the 2 rolls are independent. Let X and Y be the outcomes of the first and second rolls, respectively.

a.      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.                                                                                                                                      (10 points)

b.      What is the probability that X+Y is greater or equal to 1o?  (5 points)

3.      We have a fair eight-sided die.                                                                                     (15 points)

a.      Find the math expectation of a single roll. 

b.      Find the math expectation of the numerical sum of 4 rolls.

c.       Find the math expectation of the numerical product (i.e., multiplication)of 5 rolls.        

4.      areindependent and identically distributed random variables such that  and. What is the standard deviation of their average?  In other words, what is the standard deviation of ?                                                                                                                                                                     (5 points)

5.      If the cumulative distribution function of is  given by the function below, then find P (X < 0.80).                                                                                                                       (10 points)

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

6.      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 $35.  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 5%, respectively.  The probability distribution of Ryan’s winnings (accounting for the $5 that he paid to play) in a single game is given below. 

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

1. Find the math expectation of Ryan’s winnings for a single game.

1. Find the math expectation of Ryan’s winnings after 5 games.

1. Find the variance of Ryan’s winnings for a single  game.

1. Find the standard deviation of Ryan’s winnings for a single game.

1. Does it pay for Ryan to play this game at the fair?  Explain.

1. Find the cumulative distribution function of Ryan’s winnings for a single game and draw its graph.