Question

8 Roll a fair (standard) die until a 6 is obtained and let Y be the total number of rolls until a 6 is obtained. Also, let X the number of 1s obtained before a 6 is rolled.

(a) Find E(Y).

(b) Argue that E(X | Y = y) = 1/5 (y − 1). [Hint: The word “Binomial” should be in your answer.]

(c) Find E(X).

Answer #1

I roll a fair die until I get my first ace. Let X be the number
of rolls I need.
You roll a fair die until you get your first ace. Let Y be the
number of rolls you need.
(a) Find P( X+Y = 8)
HINT: Suppose you and I roll the same die, with me going first.
In how many ways can it happen that X+Y = 8, and what is the
probability of each of those ways?...

roll a fair die repeatedly.
a) Let X denote the number of rolls until you get at least 3
different results. Find E(X) without calculating the distribution
of X.
b) Let S denote the number of rolls until you get a repeated
result. Find E(S).

A die is rolled six times.
(a) Let X be the number the die obtained on the first roll. Find
the mean and variance of X.
(b) Let Y be the sum of the numbers obtained from the six rolls.
Find the mean and the variance of Y

a fair die was rolled repeatedly.
a) Let X denote the number of rolls until you get at least 3
different results. Find E(X) without calculating the distribution
of X.
b) Let S denote the number of rolls until you get a repeated
result. Find E(S).

A fair die is successively rolled. Let X and Y denote,
respectively, the number of rolls necessary to obtain a 5 and a 4.
Find (a) E X, (b) E[X|Y = 1] and (c) E[X|Y = 4].

A fair die is successively rolled. Let X and Y denote,
respectively, the number of rolls necessary to obtain a 5 and a 4.
Find (a) EX, (b) E[X|Y =1] and (c) E[X|Y=4].

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]

A 6-sided die rolled twice. Let E be the event "the first roll
is a 1" and F the event "the second roll is a 1".
Find the probability of showing a 1 on both rolls. Write your
answer as a reduced fraction.

Roll a die twice and let Y be the sum of the two rolls. Find the
joint pmf of (X, Y ) if X is
(a) the number on the first roll
(b) the smallest number

A fair die is rolled repeatedly. Find the expected number of
rolls until all 6 faces appear.

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