QUESTION 1 [10] It is known that 80 % of all students have their own laptops, thus in a random sample of eight students, find the following probabilities: 1.1) That exactly seven students will have their own laptops. (3) 1.2) That at least six students will have their own laptops. (4) 1.3) That at most five students will have their own laptops (4) QUESTION 2 [11] The number of accidents that occur on an assembly line have a Poisson distribution with an average of five accidents per week. 2.1) Find the probability that exactly two accidents will occur in a week. (3) 2.2) Find the probability that a particular week will be accident free. (3) 2.3) Find the probability that at least three accidents will occur in a week. (3) 2.4) If the accidents in different weeks are independent of each other, find the expected number of accidents to occur in a year. (2) QUESTION 3 [9] The time it takes a student to complete an assignment is normally distributed with mean 45 minutes and a standard deviation of 7 minutes. Find the probability that it takes the following lengths of time for a student to complete an assignment. 3.1) Between 45 and 56 minutes. (3) 3.2) Less than 33 minutes (3) 3.3) More than 29 minutes (3)

In the current tax year, suppose that 3% of the millions of individual tax returns have errors or are fraudulent. Although these errors are often well concealed, let’s suppose that a through IRS audit (done by you of course) will uncover them. If a random 100 tax returns are audited what is the probability that the IRS will uncover at most 4 fraudulent returns? Create an Excel spreadsheet that may give you an idea about this probability. Hint: Use the RAND() function for every tax return. The RAND() function generates a random number between 0 and 1. If the RAND() function gives you a number that is less than or equal to 0.03 then you can assume that the return contains error. Otherwise you can assume that the return does not contain any error. You can press F9 on your keyboard to regenerate a new instance. You may have to create enough instances to come up with a good approximation of the probability value. You can also find the exact probability using Binomial Distribution. What does it mean if an IRS auditor uncovers no more than 3 fraudulent/erroneous returns for every 100 tax returns?

Please in Excel.

2. Winning the jackpot in a particular lottery requires that you select the correct three numbers between 1 and 2828 and, in a separate​ drawing, you must also select the correct single number between 1 and 3232. Find the probability of winning the jackpot.

3. Winning the jackpot in a particular lottery requires that you select the correct five numbers between 1 and 4040 ​and, in a separate​ drawing, you must also select the correct single number between 1 and 4646. Find the probability of winning the jackpot.

The probability of winning the jackpot is ​(Type an integer or simplified​ fraction.)

4. A corporation must appoint a​ president, chief executive officer​ (CEO), chief operating officer​ (COO), and chief financial officer​ (CFO). It must also appoint a planning committee with four different members. There are 1313 qualified​candidates, and officers can also serve on the committee. Complete parts​ (a) through​ (c) below.

a. How many different ways can the officers be​ appointed?

There are_different ways to appoint the officers.

The number of hours spent studying by students on a large campus in the week before the final exams follows a normal distribution with standard deviation of 8.4 hours. A random sample of these students is taken to estimate the population mean number of hours studying.

a. How large a sample is needed to ensure that the probability that the sample mean differs from the population mean by more than 2.0 hours is less than 0.05?

b. Without doing the calculations, state whether a larger or smaller sample size compared to the sample size in part (a) would be required to guarantee that the probability of the sample mean differing from the population mean by more than 2.0 hours is less than 0.10. Explain your answer.

c. Without doing the calculations, state whether a larger or smaller sample size compared to the sample size in part (a) would be required to guarantee that the probability of the sample mean differing from the population mean by more than 1.5 hours is less than 0.05. Explain your answer.

A commuter must pass through five traffic lights on her way to work and will have to stop at each one that is red. She estimates the probability model for the number of red lights she hits (x), as shown below: x 0 1 2 3 4 5 p(x) 0.04 0.23 p 0.1 0.1 0.1 a)Find the probability that she hits at most 3 red lights. Answer to 2 decimal places. b)Find the probability that she hits at least 3 red lights. Answer to 2 decimal places. c)How many red lights she expect to hit? Answer to 2 decimal places. d)What is the standard deviation of number of red lights she hits? Answer to 3 decimal places. e)Let us consider any two consecutive days. What is the chance that she hits exactly two red lights on both days? Answer to 4 decimal places

1. Let X and Y be independent U[0, 1] random variables, so that the point (X, Y) is uniformly distributed in the unit square.

Let T = X + Y.

(a) Find P( 2Y < X ).

(b). Find the CDF F(t) of T (for all real numbers t).

HINT: For any number t, F(t) = P ( X <= t) is just the area of a part of the unit square.

(c). Find the density f(t).

REMARK: For a number t and a small dt, f(t) dt is the approximate probability that T lands in the interval [t, t+ dt ]. For example, f(0.7) *(0.01) is the approximate probability that T is between 0.7 and 0.71 . Where does (X,Y) have to land for T to be between 0.7 and 0.71 ? Now consider how the probability that T lands in [t, t + 0.01 ] changes as t increases, first from 0 to 1 and then from 1 to 2. Compare with your answer to (c).

Please include all steps. Thanks

An urn model is known in the field of probability and statistics as a useful representation of a

probabilistic problem using coloured balls in an urn. There are many different variations of an

urn model. For this problem, consider the following case:

• There is an urn with 1 red ball and 1 blue ball.

• Every time a ball is drawn (at random) from the urn, it is placed back in the urn along with

2 more balls of the same colour as the ball that was drawn, and 1 more ball of the other

colour.

Denote the random variable Xi to be the number of red balls after the i-th drawn ball (for

i = 1, 2 . . .). Note that Xi is the random variable for the number of red balls in the urn including

the three new balls added after the i-th draw.

(a) Find the probability mass function (pmf) of X2.

(b) What is the probability that the first ball drawn was red, given that there are at least 5 red

balls after the third ball is drawn.

(c) Compute E(X3) and Var(X3).

A survey of 2645 consumers by DDB Needham Worldwide of Chicago for public relations agency Porter/Novelli showed that how a company handles a crisis when at fault is one of the top influences in consumer buying decisions,with 73% claiming it is an influence. Quality of product was the number one influence, with 96% of consumers stating that quality influences their buying decisions. How a company handles complaints was number two, with 85% of consumers reporting it as an influence in their buying decisions. Suppose a random sample of 1,100 consumers is taken and each is asked which of these three factors influence their buying decisions.

Appendix A Statistical Tables

a. What is the probability that more than 810 consumers claim that how a company handles a crisis when at fault is an influence in their buying decisions?
b. What is the probability that fewer than 1,030 consumers claim that quality of product is an influence in their buying decisions?
c. What is the probability that between 81% and 84% of consumers claim that how a company handles complaints is an influence in their buying decisions?