10 red marbles and 10 blue marbles are placed into a bag. Alex mixes up the bag and randomly selects a

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.  

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

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

3. We have a fair eight-sided die.

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

, 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.  

X

-$5

$0

$10

$35

P

0.75

0.10

0.10

0.05

a. Find the math expectation of Ryan’s winningsfor a single game.

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

c. Find the varianceof Ryan’s winnings for a single game.

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

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

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