In: Statistics and Probability

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.

P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6

P(greater than 4) = P(5) + P(6) = 1/6 + 1/6 = 2/6 = 1/3 (ans)

A fair die is rolled. If the result is greater than 4, a fair
coin is flipped, and otherwise, an unfair coin with P[H] = 2/3 is
flipped. The outcome of the die roll is mapped to X, and the
outcome of the coin flip is mapped to Y , with 1 for heads and −1
for tails.
(a) Find the joint PMF.
(b) Find both marginal PMFs. Do the marginal PMFs indicate
independence?

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...

Assume that you have a fair 6 sided die with values {1, 2, 3, 4,
5, 6} and a fair 12 sided die with values {1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12}. A discrete random variable is generated by rolling
the two dice and adding the numerical results together.
(a) Create a probability mass function that captures the
probability of all possible values of this random variable. You may
use R or draw the pmf...

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...

A fair red die and a fair green die are rolled.
(a) What is the probability that the sum of the numbers is
even?
(b) What is the probability that the number on the red die is
more than the number on the green die?
(c) What is the probability that the number on the red die is
twice the number on the green die?
(d) What is the probability that the number on the red die is
different from...

Two fair coins and a fair die are tossed. Find the sample space
of the
experiment (10 pts); Find the probabilities of the following
events:
A- ”the die shows 2 or 3” (5 pts);
B- ”one of the coins is head, the other - tail, and the die shows
an odd number” (5
pts).
Are the events A and B independent? (5 pts).
Give proofs.

Suppose a single fair die is rolled.
Find the probability that it is a 2, given that it is an odd
number.
The probability is

Two coins are tossed and a (6-sided) die is rolled. Describe a
sample space(probability space), together with the probability, on
which such a situation can be modeled. Find the probability mass
function of the random variable whose value is the sum of the
number on the die and the total number of heads.

A die was rolled 300 times. The following frequencies were
recorded.
Outcome 1 2 3 4 5 6
Frequency 62 45 63 32 47 51
Do these data indicate that the die is unfair ? Use a 1% level
of significance.

A die is rolled 50 times and the following are the outputs:
6 1 3 4 2 6 3 5 1 3 6 1 6 6 3 3 6 5 2 4 1 6 5 3 1 2 5 2 1 2 4 1
4 1 5 5 6 6 2 1 1 2 5 6 5 5 6 3 1 3
What is the p-value of the chi-square test that the die is
unbiased?

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