# 1.10 red marbles and 10 blue marbles are placed into a bag. Alex mixes up the bag

Question

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)

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. (10 points)

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

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

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.

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

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 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.

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

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 varianceof 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.

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