Question

Penalty Worksheet – Due
10-19 **Expected
Value of a
Game** Name_______________________

**Instructions:**You must show your work and all
work must be organized and easy to follow. You will have to make an
appointment with me to discuss your work. You will not receive
credit if you cannot adequately explain your work to me in
person.

I pick a digit (an integer from 0 to 9, inclusive). Then, for a dollar, you get to pick three digits randomly, with replacement, by spinning a spinner (see picture below). Then the three digits you selected are compared to mine. The following describes the possible outcomes:

If none of your digits matches mine, you lose your dollar.

If exactly one of your digits matches mine, you win $1 net (your initial dollar plus one dollar from me).

If exactly two of your digits matches mine, you win $10 net.

If all three of your digits match mine, you win $100 net.

a. Let ** X**=
the number of your digits that match mine. What are the possible
values of

b. ** X**is a
binomial random variable. (Why?) If so, what are

c. Construct the probability
distribution of ** X**. (Remember, the
distribution must list the possible values of

d. Let ** W**=
your net winning for one play of the game. Describe the possible
values of

e. Construct the probability
distribution of ** W**.

f. Use the distribution of
** W**to find and interpret the expected value
(mean) of

g. Is the game "fair?” (What does it mean to describe the game as fair?) If not, determine how much you should pay to play the game in order to make it "fair." Show your work or explain your reasoning.

h. Suppose that, rather than selecting a number from 0 to 999 randomly, you are allowed to pick your own number between 0 and 999. Determine a strategy that optimizes your expected net winnings. Find the expected winnings for your strategy. (There is more than one optimal number, but only one optimal strategy.)

Answer #1

The RV represents the number of digits matcde out of 3. The probability of matching a digit is .

a )Hence is binomial Rv with .

b) The PMF of is .

Here .

c) The probability distribution of is

d) The possible values of ** W** are
.

e) The probability distribution of is

f) The expected value of is

g) The game is not fair since

*We are required to solve only 4 parts. Please post
the remaining questions as another post.
We do not get any additional amount for solving more. I have solved
7 parts.*

Penalty Worksheet – Due
10-19 Simpson’s
Paradox Name______________________________
Instructions:You must show your work and all
work must be organized and easy to follow. You will have to make an
appointment with me to discuss your work. You will not receive
credit if you cannot adequately explain your work to me in
person.
1) A manager is evaluating two baseball players based upon their
batting averages (the proportion of the time that they get a hit
when they come to bat). Russell has...

A spinner game has a wheel with the numbers 1 through 30 marked
in equally spaced slots. You pay $1 to play the game. You pick a
number from 1 to 30. If the spinner lands on your number, you win
$25. Otherwise, you win nothing. Find the expected net winnings for
this game. (Round your answer to two decimal places.)
A game costs $1 to play. A fair 5-sided die is rolled. If you
roll an even number, you...

Suppose you play a $2 scratch off lottery game where there is a
1 in 4 chance to win $2, a 1 in 10 chance to win $10, a 1 in 15
chance to win $250, and a 1 in 35 chance to win $5000.
Construct a probability distribution for x where x represents
the possible net winnings, i.e., winnings minus cost, including the
scratch off lottery games that yield no winnings.
Use the probability distribution to calculate the expected...

(Need solution for part b) You are offered to play the following
game. You roll a fair 6-sided die once and observe the result which
is shown by the random variable X. At this point, you can stop the
game and win X dollars. Or, you can also choose to discard the X
dollars you win in the first roll, and roll the die for a second
time to observe the value Y . In this case, you will win...

In a game of “Chuck a luck” a player bets on one of the numbers
1 to 6. Three dice are then rolled and if the number bet by the
player appears i times (where i equals to 1, 2 or 3) the player
then wins i units. On the other hand if the number bet by the
player does not appear on any of the dice the player loses 1 unit.
If x is the players’ winnings in the...

4. (1 point) Consider the following pmf:
x = -2 0 3 5
P(X = x) 0.34 0.07 ? 0.21
What is the
E[
X 1 + X
]
5. (1 point) Screws produced by IKEA will be defective with
probability 0.01 independently of each other. What is the expected
number of screws to be defective in any given pack? Hint: you
should be able to recognize this distribution and use the result
from the Exercises at the end of...

1.
Create a PDF table and calculate expected
value.
A friend offers you a game to play where you pay him $10. You
roll a fair 6-sided die. If the roll of a comes up as 1, 2, 3 he
pays you $5. If the roll is 4 or 5 he pays you $7 and if it is a 6
he pays you $20.
In words, define the random variable X. ?
Construct a PDF table.
If you play this...

7. Imagine you are in a game show. There are 10
prizes hidden on a game board with 100 spaces. One prize is worth
$50, three are worth $20, and another six are worth $10. You have
to pay $5 to the host if your choice is not correct. Let the random
variable x be the winning.
Complete the following probability distribution. (Show the
probability in fraction format and explain your work)
x
P(x)
-$5
$10
$20
$50
What is...

14. Imagine you are in a game show. There are 10 prizes hidden
on a game board with 100 spaces. One prize is worth $50, three are
worth $20, and another six are worth $10. You have to pay $5 to the
host if your choice is not correct. Let the random variable x be
the winning.
a. Complete the following probability distribution. (Show the
probability in fraction format and explain your work)
x
P(x)
-$5
$10
$20
$50
b....

Here are the basic rules of Club Keno:
You choose how many numbers you will pick. You can pick
anywhere from 1 to 10 different numbers.
Pick your numbers between 1 and 80.
A drawing is held in which 20 numbers are picked.
Depending on how many of your numbers come up in the drawing,
you win various amounts of money.
In a previous assignment we saw the number of ways to choose
exactly x correct from a total of...

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