1. 10 red marbles and 10 blue marbles are placed into a bag. Alex mixes up thebag and randomly selects a marble. He continues to do so, replacing themarble after each selection, until a red marble is selected.(10points)a. What is the probability that the first time that a red marble is pulledis on Alex’s 6th try?b. On average, how many marbles will Alex have to pull in order to get ared 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 tablethat contains each unique possible value of X+Y (each value onlylisted 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.a. Find the math expectation of a single roll.(15 points)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.X 1 , X 2 ,… X 121 are independent and identically distributed random variablessuch thatE ( X i ) =3andVar ( X i )=25 . What is the standard deviation oftheir average? In other words, what is the standard deviation ofX X 1+ X 2 +…+ X 121?121(5 points)5. If the cumulative distribution function ofthen find P (X < 0.80)., if x ≤ 0F ( x) 0x 3F ( x) 7 7x2, if 0F ( x) 1x≤1, if x > 1Xis given by the function below,(10 points) 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 themedium bottle, you win $10. If you get a “ringer” on the large bottle, you getyour $5 fee back (that is, you break even). If you miss, you are out the $5 youpaid to play. Ryan is a good shot and his probability of getting a ringer on thesmall, medium, and large bottles is 10%, 10%, and 5%, respectively. Theprobability distribution of Ryan’s winnings (accounting for the $5 that he paidto play) in a single game is given below.(5 points each for parts a-e and 20 points for part f)XP-$50.75$00.10$100.10$350.05a. Find the math expectation of Ryan’s winnings for a single game.b. Find the math expectation of Ryan’s winnings after 5 games.c. Find the variance of 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 asingle game and draw its graph.