In: Statistics and Probability

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

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

P(1) = 1/4

P(not 1) = 1-1/4 = 3/4

probability(outcome 1 is first observed after 5 rolls)

= P(1 doesn't come in 5 rolls)

= P(not 1)^5

= (3/4)^5

= **0.2373**

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

The first number (either 1 or 2) will come up with probability 2/4 on each roll, and the number of rolls needed follows a geometric distribution, so 4/2 = 2 rolls are expected.

Then the remaining number comes up with probability 1/4, and similar to the first number, the reciprocal number of rolls are expected, or 4/1 = 4 rolls.

total rolls = 2+4 = 6

**Thus 6 rolls are expected on average.**

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

expected no. of rolls for event to occur x times = x / (p(event))

P(3) = 1/4

expected no. of rolls for outcome 3 to occur 3 times = 3 / (1/4)
= **12**

**expected no. of rolls for outcome 3 to occur 3 times =
12**

(d) probability that the outcome 3 is observed exactly three times in 10 rolls given that it is first observed after 5 rolls.

x=no. of 3

p=1/4

P(3 observed 3 times in 10 rolls | 3 not observed in first 5 rolls)

= P(3 observed 3 times in the 5 rolls afterr first 5 rolls)

= 5C3 * (1/4)^3 * (1 - 1/4)^(5-3)

= **0.0879**

(e) probability that the outcome 3 is first observed after 5 rolls
given that it is observed exactly three times in 10 rolls

x=no. of 3

p=1/4

P(3 observed 3 times in 10 rolls) = 10C3 * (1/4)^3 * (1 - (1/4))^(10-3)

= **0.2503**

P(3 is first observed after 5 rolls)

= P(3 doesn't come in 5 rolls)

= P(not 3)^5

= (1 - 1/4)^5

= **0.2373**

P(3 observed 3 times in 10 rolls | 3 is first observed after 5 rolls)

= **0.0879 {from previous part}**

P( 3 is first observed after 5 rolls | 3 observed 3 times in 10 rolls)

= P(3 observed 3 times in 10 rolls | 3 is first observed after 5 rolls)*P(3 is first observed after 5 rolls) / P(3 observed 3 times in 10 rolls)

= 0.0879 * 0.2373 / 0.2503

= 0.0833

**P( 3 is first observed after 5 rolls | 3 observed 3
times in 10 rolls) = 0.0833**

**(please UPVOTE)**

Let X equal the outcome (1, 2 , 3 or 4) when a fair four-sided
die is rolled; let Y equal the outcome (1, 2, 3, 4, 5 or 6) when a
fair six-sided die is rolled. Let W=X+Y.
a. What is the pdf of W?
b What is E(W)?

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...

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.

17#13
Suppose we roll a fair six-sided die and sum the values obtained
on each roll, stopping once our sum exceeds 289. Approximate the
probability that at least 76 rolls are needed to get this sum.

Suppose you are rolling a fair four-sided die and a fair
six-sided die and you are counting the number of ones that come
up.
a) Distinguish between the outcomes and events.
b) What is the probability that both die roll ones?
c) What is the probability that exactly one die rolls a one?
d) What is the probability that neither die rolls a one?
e) What is the expected number of ones?
f) If you did this 1000 times, approximately...

Suppose a 9-sided and a 4-sided die are rolled. Find these
probabilities. a. P(roll sum of less than 5 or roll doubles)
b. P(roll a sum greater than 6 for the first time on the sixth
roll)

Suppose we roll a fair 6 sided die with the numbers [1,6]
written on them. After the first die roll we roll the die ? times
where ? is the number on the first die roll. The number of points
you score is the sum of the face-values on all die rolls (including
the first). What is the expected number of points you will
score?

Find the conditional probability, in a single roll of two fair
6-sided dice, that neither die is a
three,given that the sum is greater than 7.

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.

ADVERTISEMENT

ADVERTISEMENT

Latest Questions

- Let P2 be the vector space of all polynomials of degree less than or equal to...
- Develop a python program to create a quiz with limited time and attempts!!! Add comments and...
- Quantitative noninvasive techniques are needed for routinely assessing symptoms of peripheral neuropathies, such as carpal tunnel...
- The Blending Department of Luongo Company has the following cost and production data for the month...
- A suspension of calcium carbonate particles in water flows through a pipe. An engineer was asked...
- O’Brien Company manufactures and sells one product. The following information pertains to each of the company’s...
- 1. Find the Legendre polynomial PL(x) for L = 3,4,5,6 where the polynomian is the series...

ADVERTISEMENT