Question

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]

Answer #1

The probability of getting an 8 in two dice throws is first computed here as:

Total outcomes = 6*6 = 36

Sum of 8 can be obtained as: 2 + 6, 3 + 5, 4 + 4, 5 + 3 or 6 + 2
that is 5 cases. Therefore the probability of getting a sum of 8 is
computed here as: 5/36

The expected number of rolls until we get two consecutive sums of 8 be X. Also when one 8 has already occurred let the same probability be Y.

Then, we have here:

X = (5/36)*(Y+ 1) + (31/36)*(1 + X)

This is because there is a 5/36 probability that an 8 will come in
which case there would be Y expected more rolls to get two
consecutive 8s.

(5/36) X = 1 + (5/36)Y

Also, for Y we have here:

Y = (5/36)*1 + (31/36)*(1 + X)

Y = 1 + (31/36)X

Putting this in the previous equation, we get here:

(5/36) X = 1 + (5/36)(1 + (31/36)X )

(5/36)X = 1 + (5/36) + (155/36^{2})X

X = 59

**Therefore 59 is the required expected number of tosses
required here.**

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 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]

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.

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?...

Please follow the comment.
2. Roll two fair dice repeatedly. If the sum is ≥ 10, then you
win.
(a) What is the probability that you start by winning 3 times in
a row?
(b)What is the probability that after rolling the pair of dice 5
times you win exactly 3 times?
(c) What is the probability that the first time you win is
before the tenth roll (of the pair), but after the fifth?

Using Python :
consider the following experiment:
You roll a pair of fair dice and calculate the sum of the
faces.
You are interested in the number of rolls it takes until you get
a sum of "7". The first time you get a "7" the experiment is
considered a "success". You record the number of rolls and you stop
the experiment. You repeat the experiment N=100,000 times. Each
time you keep track of the number of rolls it takes...

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.

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