Question

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?

(b) Find P(X+Y >= 8)

(c) Find P(X = Y).

(d) Find P(X > Y+2).

HINT: What are the values of P(X >Y) and P(X > Y+2 | X>Y) ? Or, for a more grind-it-out solution, sum the probabilities of all (x,y) values for which the event X > Y+2 happens.

Answer #1

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).

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 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 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

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...

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

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]

You flip a fair coin. If the coin lands heads, you roll a fair
six-sided die 100 times. If the coin lands tails, you roll the die
101 times. Let X be 1 if the coin lands heads and 0 if the coin
lands tails. Let Y be the total number of times that you roll a 6.
Find P (X=1|Y =15) /P (X=0|Y =15) .

Roll a fair four-sided die twice. Let X be the sum of the two
rolls, and let Y be the larger of the two rolls (or the common
value if a tie).
a) Find E(X|Y = 4)
b) Find the distribution of the random variable E(X|Y )
c) Find E(E(X|Y )). What does this represent?
d) Find E(XY |Y = 4)
e) Find the distribution of the random variable E(XY |Y )
f) Explain why E(XY |Y ) = Y...

Roll a fair 6-sided die repeatedly and letY1,Y2,...be the
resulting numbers. Let Xn=|{Y1,Y2,...,Yn}|be the number of values
we have seen in the first n rolls for n≥1 and setX0= 0.Xn is a
Markov chain.(a) Find its transition probability.(b) Let T=
min{n:Xn= 6}be the number of trials we need to see all 6 numbers at
least once. Find E[T]. Please explain how/why

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