1. Suppose four distinct, fair coins are tossed. Let the random variable X be the number of heads. Write the probability mass function f(x). Graph f(x).

2.  For the probability mass function obtained, what is the cumulative distribution function F(x)? Graph F(x).

3. Find the mean (expected value) of the random variable X given in part 1

4. Find the variance of the random variable X given in part 1.

1. Complaints about an Internet brokerage firm occur at a rate of 3 per day. The number of complaints appears to be Poisson distributed.

A. Find the probability that the firm receives 3 or more complaints in a 2-day period.

2. In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 13% of voters are Independent. A survey asked 15 people to identify themselves as Democrat, Republican, or Independent.

A. What is the probability that more than 3 people are Independent?

The following is the transition probability matrix of a Markov chain with states 1, 2, 3, 4 P

If Xnot = 1
(a) find the probability that state 3 is entered before state 4;

(b) find the mean number of transitions until either state 3 or state 4 is entered.

Suppose that in the certain country the proportion of people with red hair is 29%. Find the following probabilities if 37 people are randomly selected from the populattion of this country. Round all probabilities to four decimals.

(a) The probability that exactly 6 of the people have red hair

(b) The probability that at least 6 of the people have red hair

(c) Out of the sample of 37 people, it would be unusual to have more than people with red hair. Express your answer as a whole number.

Question 1 - Binomials

Eighty percent of the students applying to a university are accepted. Using the binomial probability tables or Excel, what is the probability that among the next 15 applicants:

1. Exactly 6 will be accepted?
2. At least 10 will be accepted?
3. Exactly 3 will be rejected?
4. Eight or more will be accepted?
5. Determine the expected number of accepted students. Remember you are talking about people. Work is required.
6. Compute the standard deviation.- Make sure you have looked at page 245. Work is required.

I have no strong background in Probability, please, present to me an easy to understand solution to this problem with detail explanation. Thank you.

An urn contains three white, six red, and five black balls. Six of these balls are randomly selected from the urn. Let X and Y denote
respectively the number of white and black balls selected. Compute the conditional probability mass function of X given that Y = 3. Also compute E[X|Y = 1]

Consider an experiment where you toss a coin as often as necessary to turn up one head.Suppose that the probability of having a tail is p(obviously probability of a head is 1−p).Assume independence between tosses.a) State the sample space.

b) Let X be the number of tosses needed to get one head. What is the support (possible values ofX)?

c) FindP(X= 1),P(X= 2) andP(X= 3).

d) Deduce the pmf of X from part c).

Suppose that a sample space consists of ? equally likely outcomes. Select all of the statements that must be true.

a. Each outcome in the sample space has equal probability of occurring.

b. Any two events in the sample space have equal probablity of occurring.

c. The probability of any event occurring is the number of ways the event can occur divided by ?.

d. Probabilities can be assigned to outcomes in any manner as long as the sum of probabilities of all outcomes in the sample space is 1.

Find the indicated probabilities using the geometric​ distribution, the Poisson​ distribution, or the binomial distribution. Then determine if the events are unusual. If​ convenient, use the appropriate probability table or technology to find the probabilities.

The mean number of oil tankers at a port city is

1313

per day. The port has facilities to handle up to

1717

oil tankers in a day. Find the probability that on a given​ day, (a)

thirteenthirteen

oil tankers will​ arrive, (b) at most three oil tankers will​ arrive, and​ (c) too many oil tankers will arrive.

Here is a simple probability model for multiple-choice tests. Suppose that each student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) The correctness of answers to different questions are independent. Jodi is a good student for whom p = 0.75.

(a) Use the Normal approximation to find the probability that Jodi scores 70% or lower on a 100-question test. (Round your answer to four decimal places.)

=


(b) If the test contains 250 questions, what is the probability that Jodi will score 70% or lower? (Use the normal approximation. Round your answer to four decimal places.)
=

(c) How many questions must the test contain in order to reduce the standard deviation of Jodi's proportion of correct answers to half its value for a 100-item test?
= questions

(d) Laura is a weaker student for whom p = 0.7. Does the answer you gave in (c) for standard deviation of Jodi's score apply to Laura's standard deviation also? choose one A or B

A.Yes, the smaller p for Laura has no effect on the relationship between the number of questions and the standard deviation.

B.No, the smaller p for Laura alters the relationship between the number of questions and the standard deviation.