Question

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).

Answer #1

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).

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]

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?...

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]

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].

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).

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

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

Consider an experiment where a fair die is rolled repeatedly
until the first time a 3 is observed.
i) What is the sample space for this experiment? What is the
probability that the die turns up a 3 after i rolls?
ii) What is the expected number of times we roll the die?
iii) Let E be the event that the first time a 3 turns up is after
an even number of rolls. What set of outcomes belong to...

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