Question

Assume we roll a fair four-sided die marked with 1, 2, 3 and
4.

(a) Find the probability that the outcome 1 is first observed after
5 rolls.

(b) Find the expected number of rolls until outcomes 1 and 2 are
both observed.

(c) Find the expected number of rolls until the outcome 3 is
observed three times.

(d) Find the probability that the outcome 3 is observed exactly
three times in 10 rolls

given that it is first observed after 5 rolls.

(e) Find the probability that the outcome 3 is first observed after
5 rolls given that

it is observed exactly three times in 10 rolls

Answer #1

Assume we roll a fair four-sided die marked with 1, 2, 3 and
4.
(a) Find the probability that the outcome 1 is first observed after
5 rolls.
(b) Find the expected number of rolls until outcomes 1 and 2 are
both observed.
(c) Find the expected number of rolls until the outcome 3 is
observed three times.
(d) Find the probability that the outcome 3 is observed exactly
three times in 10 rolls
given that it is first observed...

1. Suppose you have a fair 6-sided die with the numbers 1
through 6 on the sides and a fair 5-sided die with the numbers 1
through 5 on the sides. What is the probability that a roll of the
six-sided die will produce a value larger than the roll of the
five-sided die?
2. What is the expected number of rolls until a fair five-sided
die rolls a 3? Justify your answer briefly.

Assume that a fair
six-sided die is rolled 9 times, and the roll is called a success
if the result is in {1,2}{1,2}.
What is the probability that there are exactly 4 successes or
exactly 4 failures in the 9 rolls?

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...

You have a 4 sided die with sides marked A, B, C and
D. You roll the die 100 times. Find the
probablilty that you get A at least 92 % of the time.
Your friend also rolls the die from problem three 100 times. He
rolled more A’s than 70 % of all people who could roll this die 100
times. What percentage of the time did he roll and
A???

1. You have a 4 sided die with sides marked A, B, C and D. You
roll the die 100 times. Find the probability that you get A at
least 92 % of the time.
2. Your friend also rolls the die from problem 1, 100 times. He
rolled more A’s than 70 % of all people who could roll this die 100
times. What percentage of the time did he roll and A???

You roll a fair 6 sided die 5 times. Let Xi be the
number of times an i was rolled for i = 1, 2, . . . , 6.
(a) What is E[X1]?
(b) What is Cov(X1, X2)?
(c) Given that X1 = 2, what is the probability the
first roll is a 1?
(d) Given that X1 = 2, what is the conditional probability mass
function of, pX2|X1 (x2|2), of
X2?
(e) What is E[X2|X1]

Suppose you roll a single fair die 10 times (all rolls are
mutually independent). What is the probability that you roll a 5 in
exactly 3 of the 10 rolls?

A fair six-sided die has two sides painted red, 3 sides painted
blue and one side painted yellow.
The die is rolled and the color of the
top side is recorded.
List all possible outcomes of this random experiment
Are the outcomes equally likely? Explain
Make a probability distribution table for the random variable
X: color of the top side
2. If a pair of dice painted the same way as in problem 1 is
rolled, find the probability...

A fair 4-sided die is rolled 7 times.
(a)
Find the probability that the side 1 comes up exactly 3
times.
(b)
Find the probability that there is at least one side that comes
up exactly 3 times.

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