Question

A fair die is rolled three times. Let X denote the number of different faces showing, X = 1, 2, 3. Find E(X). Give a good explanation please

Answer #1

Total number of outcomes for three times rolled dice =
6^{3}

Let X be the number of different faces.

P(X = 1) = P(all faces are same)

[ Possible events are { (1,1,1),(2,2,2) ....,(6,6,6) } = 6 outcomes ]

= 6 / 6^{3}

= 1 / 36

P(X = 2) = P(2 different faces)

[ One number is fixed and other two numbers are different then

for first number there are 6 choices and second number there are 5 choices) ]

= ^{3}C_{2} * 1 * 6 * 5 / 6^{3}

= 3 * 6 * 5 / 6^{3}

= 15 / 36

P(3 differnt faces) = ^{3}C_{3} * 6 * 5 * 4 /
6^{3}

= 20 / 36

E(X) = X * P(X)

= 1 * 1/36 + 2 * 15/36 + 3 * 20/36

= **91/36 ( = 2.5278)**

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].

A fair die is rolled repeatedly. Let X be the random variable
for the number of times a fair die is rolled before a six appears.
Find E[X].

Suppose a fair die is tossed three times. Let X be the smallest
of the three faces that appear.
a)Find pX(k).
b)Let Y be the same of the three faces that appear, Find
pY(k).
please explain how to get that answer

Three fair dice are rolled. Let X be the total number of spots
showing, that is the sum of the results of the three rolls.
a) Find the probabilities P(X= 3), P(X= 4), P(X= 17), P(X=
18).
b) Find the probability P(X≥ 11).

A fair six-sided die is rolled 10 independent times. Let X be
the number of ones and Y the number of twos.
(a) (3 pts) What is the joint pmf of X and Y?
(b) (3 pts) Find the conditional pmf of X, given Y = y.
(c) (3 pts) Given that X = 3, how is Y distributed
conditionally?
(d) (3 pts) Determine E(Y |X = 3).
(e) (3 pts) Compute E(X2 − 4XY + Y2).

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

1) A 10-sided die is rolled infinitely many times. Let X be the
number of rolls up to and including the first roll that comes up 2.
What is Var(X)?
Answer: 90.0
2) A 14-sided die is rolled infinitely many times. Let X be the
sum of the first 75 rolls. What is Var(X)?
Answer: 1218.75
3) A 17-sided die is rolled infinitely many times. Let X be the
average of the first 61 die rolls. What is Var(X)?
Answer:...

A fair die is rolled 1000 times. Let A be the event that the
number of 6’s is in the interval[150,200], and B the event that the
number of 5’s is exactly 200. (a) Approximate P(A).(b) Approximate
P(A|B).

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