2 red marbles and 8 blue marbles are placed into a bag
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question.docx1. 2 red marbles and 8 blue marbles are placed into a bag. Bob 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. (12 points)
a. What is the probability that the first time that a red marble is pulled is on Bob’s 8th try?
b. On average, how many marbles will Bob have to pull in order to get a red marble? (Hint: use math expectation)
2. Let a fair die be rolled n times. Let’s make an assumption that all rolls of a die are independent. If X and Y are the outcomes of any two of n trials, 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. (15 points)
3. We have a fair ten-sideddie. (15 points)
a. Find the math expectation of a single roll.
b. Find the math expectation of the numerical sum of 5 rolls.
c. Find the math expectation of the numerical product (i.e., multiplication) of 7 rolls.
4.
5.
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 $50. 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 20%, respectively. The probability distribution of Ryan’s winnings in a single game is given below.
|
X |
-$5 |
$0 |
$10 |
$50 |
|
P |
0.60 |
0.20 |
0.10 |
0.10 |
a. Find the math expectation of Ryan’s winningsfor a single game.
b. Find the math expectation of Ryan’s winnings after 3 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.
(5 points each for parts a-e and 20 points for part f)
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