In: Statistics and Probability

A three-sided fair die with faces numbered 1, 2 and 3 is rolled twice. List the sample space. S =

b.{ List the following events and their probabilities. Write probabilities in non-reduced fractional form A = rolling doubles = { P(A)= / B = rolling a sum of 4 = { P(B)= / C = rolling a sum of 5 = { P(C)=

C. Are the events A and B mutually exclusive? If yes, why? If not, why not?

D.Are the events B and C mutually exclusive? If yes, why? If not, why not?

E. Find the following probabilities: P(A and B)= / P(B or C)= / P(B and C)=

**Question (a)**

A three-sided fair die with faces numbered 1, 2 and 3 is rolled twice

Sample Space S = { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) }

**Question (b)**

Total number of outcomes possible = 9

A = rolling doubles

Outcomes where doubles will roll are (1,1), (2,2), (3,3)

So Number of outcomes for A = 3

So **P(A) = 3/9**

B = rolling a sum of 4

Outcomes where sum will be 4 on both rolls are (1,3), (2,2), (3,1)

So Number of outcomes for B = 3

So **P(B) = 3/9**

C = rolling a sum of 5

Outcomes where sum will be 5 on both rolls are (2,3), (3,2)

So Number of outcomes for C = 2

So **P(C) = 2/9**

**Question (c)**

Are the events A and B mutually exclusive?

Two events A and B are said to be mutually exclusive if they don't have any common outcomes between them

Here A and B have a common outcome which is (2,2)

So events A and B are not mutually exclusive

**Question (d)**

Are the events B and C mutually exclusive?

Two events B and C are said to be mutually exclusive if they don't have any common outcomes between them

Here B and C do not have any common outcome between them

So events B and C are mutually exclusive

**Question (e)**

P(A and B)

The number of common outcomes between A and B is 1 which is (2,2)

So **P(A and B) = 1/9**

P(B or C)

P(B or C) = P(B) + P(C) - P(B and C)

B and C does not have any common outcomes between them, So P(B and C) = 0

P(B or C) = P(B) + P(C)

= 3/9 + 2/9

= 5/9

**P(B or C) = 5/9**

B and C does not have any common outcomes between them, So
**P(B and C) = 0**

2 fair four-sided dice with faces numbered 1, 2, 3, 4 are
rolled. Random variable D is the nonnegative difference |X2 −X1| of
the numbers rolled on the 2 dice and Y is a Bernoulli random
variable such that Y = 0 if the sum X1 +X2 is even and and Y = 1 if
the sum is odd.
a. Find joint probability mass function of D and Y.
b. Are D and Y dependent or independent? How do you...

A fair four sided die has two faes numbered 0 and two faces
numbered 2. Another fair four sided die has its faces numbered
0,1,4, and 5. The two dice are rolled. Let X and Y be the
respective outcomes of the roll. Let W = X + Y.
(a) Determine the pmf of W.
(b) Draw a probability histogram of the pmf of W

If a fair coin is flipped twice and a standard 6 sided die is
rolled once, what is the likelihood of getting two 'heads' on the
coin and a '1' on the die? â€‹Express your answer as a percent
rounded to the tenth place

Consider rolling both a fair four-sided die numbered 1-4 and a
fair six-sided die numbered 1-6 together. After rolling both dice,
let X denote the number appearing on the foursided die and Y the
number appearing on the six-sided die. Define W = X +Y . Assume X
and Y are independent.
(a) Find the moment generating function for W.
(b) Use the moment generating function technique to find the
expectation.
(c) Use the moment generating function technique to find...

Consider a fair four-sided die numbered 1-4 and a fair six-sided
die numbered 1-6, where X is the number appearing on the
four-sided die and Y is the number appearing on the
six-sided die. Define W=X+Y when they are rolled
together. Assuming X and Y are
independent, (a) find the moment generating function for
W, (b) the expectation E(W), (c) and the variance
Var(W). Use the moment generating function technique to
find the expectation and variance.

Example 4: A fair six-sided die is rolled six times. If
the face numbered k is the outcome on roll k for k = 1, 2, 3, 4, 5,
6 we say that a match has occurred. The experiment is called a
success if at least one match occurs during the six trials.
Otherwise, the experiment is called a failure. The outcome space is
O = {success, failure}. Let event A = {success}. Which value has
P(A)?
**This question has...

a fair 6-sided die is rolled three times. What is the
probability that all three rolls are 2? Round the answer to four
decimal places

1. The experiment of rolling a fair six-sided
die twice and looking at the values of the faces that are facing
up, has the following sample space.
For example, the result (1,2) implies that the face that is up
from the first die shows the value 1 and the value of the face that
is up from the second die is 2.
(1,1)
(1,2)
(1,3)
(1,4)
(1,5) (1,6)
(2,1)
(2,2)
(2,3)
(2,4)
(2,5) (2,6)
(3,1)
(3,2)
(3,3)
(3,4)
(3,5) (3,6)...

Assume that a fair die is rolled. The sample space is
, 1, 2, 3, 4, 56
, and all the outcomes are equally likely. Find
P
Greater than 4
. Write your answer as a fraction or whole number.
Assume that a fair die is rolled. The sample space is
, 1, 2, 3, 4, 56
, and all the outcomes are equally likely. Find
P
Greater than 4
. Write your answer as a fraction or whole number.

The experiment of rolling a fair six-sided die twice and looking
at the values of the faces that are facing up, has the following
sample space.
For example, the result (1,2) implies that the face that is up
from the first die shows the value 1 and the value of the face that
is up from the second die is 2.
sample space of tossing 2 die
A pair of dice is thrown.
Let X = the number of multiples...

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