Question

If you roll a die, you get one of the following numbers: 1, 2, 3, 4, 5, 6. Each possibility occurs with equal probability of 1/6. The expected value of a dice roll is E(D)= 3.5 and the variance of a dice roll is Var(X) = 2.917.

a) Suppose you roll a die and then add 1 to the roll to get a new random variable taking one of the following numbers: 2,3,4,5,6,7. What is the variance of this new random variable?

b) Suppose you roll a die and then multiply the roll by 2 to get a new random variable taking one of the following numbers: 2, 4, 6, 8, 10, 12. What is the variance of this new random variable?

c) Suppose you roll two dice and then take the sum of the two dice to get a number between 2 and 12. What is the variance of this sum random variable?

Answer #1

I have a 6 sided die with the numbers 1, 2, 3, 3, 4, 5 on it. I
also have a hat filled with numbers. When I pick a number out of
the hat, I get a number between 1 and 10. The hat number has a mean
of 2 and a standard deviation of 2.5. I make a new random variable
by combining these two previous random variables. The new random
variable is made by taking the number I...

You roll one red die and one green die. Define the random
variables X and Y as follows: X = the absolute value of difference
of the number on the red die and the number on the green die. Y =
the sum of the number of dice that show the number 1.
For example, if the red and green dice show the numbers 3 and 6,
then X=3 and Y =0.
(a) Make a table showing the joint probability...

There are four dice with varying numbers on each face:
Die A. has 1 2 4 5 6 7
Die B. has 4 4 4 4 4 4
Die C. has 2 6 2 6 2 2
Die D. has 3 5 3 5 3 5
You and three friends play a game where you each roll one of the
dice and the highest number wins.
You get first pick from the dice. Question: Which die should you
choose in...

a. Roll a dice, X=the number obtained. Calculate E(X), Var(X).
Use two expressions to calculate variance.
b. Two fair dice are tossed, and the face on each die is
observed. Y=sum of the numbers obtained in 2 rolls of a dice.
Calculate E(Y), Var(Y).
c. Roll the dice 3 times, Z=sum of the numbers obtained in 3
rolls of a dice. Calculate E(Z), Var(Z) from the result of part a
and b.

Suppose you roll two six-sided dice at once, one yellow and the
other green.
Find the probability that: (1) The yellow die shows 3 and the
green shows
5. (2) The yellow die is even and the green shows 1. (3) The sum
of the
two numbers shown is 6.

You roll a six-sided die repeatedly until you roll a one. Let X
be the random number of times you roll the dice. Find the following
expectation:
E[(1/2)^X]

QUESTION 5: A 4-sided die is used for some games. This die can
only land on the numbers 1, 2, 3, or 4. All these outcomes are
equally likely. Suppose we roll two 4-sided dice.
a. (3 pts) Write out the sample space (the set of all the ways
in which the two dice could land).
b. (2 pts) What is the probability that at least one of the dice
shows a 3?
c. (3 pts) Suppose we compute the...

If you roll a 4-sided die and a 6-sided die at the same time and
then add the result on each of the dice, how many different ways
can you get a number greater than 7?

Question I: You roll 2 fair dice, one red and one
green.
a) What is the probability that the red die does land on
4? does not land on 4?
b) What is the probability that either die lands on 4?
that neither die lands on 4?
c) What is the probability that the sum of the dice is
2? 3? 4? 5? 6? 7? 8? 9? 10? 11? 12?

4 fair 10-sided dice are rolled.
(a)
Find the conditional probability that at least one die lands on
3 given that all 4 dice land on different numbers.
(b)
True or False: If X is the sum of the 4 numbers from
one roll, and Y is the maximum of the 4 numbers from one
roll, then X and Y are independent random
variables.

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