Assignment 5 | Complete Solution

Assignment 5

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

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

    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.
    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.                                    

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

     We have a fair eight-sided die.                        
    Find the math expectation of a single roll.   

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

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

    X_1,X_2,… X_121 are independent and identically distributed random variables such that E(X_i )=3 and  Var(X_i )=25. What is the standard deviation of their average?  In other words, what is the standard deviation of  (X_1+X_2+ …+X_121)/121?                                            

    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

    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

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

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

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

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

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

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