Question

Consider some 8-sided dice. Roll two of these dice. Let X be the minimum of the two values that appear. Let Y denote the maximum.

a) Find the joint mass p_{X,Y}(x,y).

b) Compute p_{X}_{│}_{Y}(x│y) in all
cases. Express your final answer in terms of a table.

Answer #1

Consider a game where two 6-sided dice are rolled.
- Let X be the minimum of the two dice
- Let Y be the sum of the two dice
- Let Z be the first die minus the second die.
Write out the distributions of X, Y, and Z, respectively.

Roll two six sided dice. Let Y be the random variable the
represents the product of the two dice.
Define W = 4Y + 2 and find E(W).

Two identical fair 6-sided dice are rolled
simultaneously. Each die that shows a number less than or equal to
4 is rolled once again. Let X be the number of dice that show a
number less than or equal to 4 on the first roll, and let Y be the
total number of dice that show a number greater than 4 at the
end.
(a) Find the joint PMF of X and Y . (Show your final
answer in a...

Consider rolling two fair six-sided dice.
a) Given that the roll resulted in sum of 8, find the
conditional probability that first die roll is 6.
b) Given that the roll resulted in sum of 4 or less, find the
conditional probability that doubles are rolled.
c) Given that the two dice land on different numbers, find the
conditional probability that at least one die is a 6.

You roll a pair of fair dice repeatedly. Let X denote the number
of rolls until you get two consecutive sums of 8(roll two 8 in a
row). Find E[X]

Two fair six-sided dice are rolled once. Let (X, Y) denote the
pair of outcomes of the two rolls.
a) Find the probability that the two rolls result in the same
outcomes.
b) Find the probability that the face of at least one of the
dice is 4.
c) Find the probability that the sum of the dice is greater than
6.
d) Given that X less than or equal to 4 find the probability
that Y > X.

Roll two dice (one red and one white). Denote their outcomes as
X1 and X2. Let T = X1+X2 denote the total, let X1 W X2 denote the
maximum and let X1 V X2 denote the minimum. Find the following
probabilities: (a) P(X1 ≥ 3|X2 ≤ 4) (b) P(T is prime) (c) P(T ≤
8|X1 W X2 = 5) (d) P(X1 V X2 ≤ 5|T ≥ 8) (e) P(X1 W X2 ≥ 3|X1 W X2 ≤
3)

Let X be the outcome of a roll of a 6-sided die, with possible
values 1,2,3,4,5,6, and let Y be the outcome of a roll of a
10-sided die, with possible values 1,2,3,4,5,6,7,8,9, and 10.
Explain briefly why X and Y are not identically distributed.

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]

5 fair 8-sided dice are rolled.
(a)
[3 marks] Find the conditional probability that at least one
die lands on 3 given that all 5 dice land on different
numbers.
(b)
[2 marks] True or False: If X is the minimum of the 5
numbers from one roll, and Y is the sum of the 5 numbers
from one roll, then X and Y are independent
random variables.

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