“Please answer all the questions and provide rationale for the answer for each question using your own words and show your calculations to get full credit”. 1. Which of the following research question we can use case-control study to answer? [ONE POINT] a. What is the past mortality or morbidity trends that can help estimates of the occurrence of disease in the future? b. What is the relative frequency of the characteristic or exposure under study when you compare histories and other information from a group of cases and from a comparison group? c. What is the...

An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of them are white and 2 are black, we stop. If not, we replace the balls in the urn and again randomly select 4 balls. This continues until exactly 2 of the 4 chosen are white. What is the probability that we shall make exactly n selections?

Two athletic teams play a series of games; the first team to win 4 games is declared the overall winner. Suppose that one of the teams is stronger than the other and wins each game with probability .6, independently of the outcomes of the other games. Find the probability, for i = 4, 5, 6, 7, that the stronger team wins the series in exactly i games. Compare the probability that the stronger team wins with the probability that it would win a 2-out of-3 series.

Consider a roulette wheel consisting of 38 numbers 1 through 36, 0, and double 0. If Smith always bets that the outcome will be one of the numbers 1 through 12, what is the probability that (a) Smith will lose his first 5 bets; (b) his first win will occur on his fourth bet?

If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1/100, what is the (approximate) probability that you will win a prize (a) at least once? (b) exactly once? (c) at least twice?

The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the probability that there will be (a) at least 2 such accidents in the next month; (b) at most 1 accident in the next month? Explain your reasoning!

The expected number of typographical errors on a page of a certain magazine is .2. What is the probability that the next page you read contains (a) 0 and (b) 2 or more typographical errors? Explain your reasoning!

It is known that diskettes produced by a certain company will be defective with probability .01, independently of each other. The company sells the diskettes in packages of size 10 and offers a money-back guarantee that at most 1 of the 10 diskettes in the package will be defective. The guarantee is that the customer can return the entire package of diskettes if he or she finds more than one defective diskette in it. If someone buys 3 packages, what is the probability that he or she will return exactly 1 of them?

A satellite system consists of n components and functions on any given day if at least k of the n components function on that day. On a rainy day each of the components independently functions with probability p1, whereas on a dry day they each independently function with probability p2. If the probability of rain tomorrow is α, what is the probability that the satellite system will function?

A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability .2. Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit 00000 instead of 0 and 11111 instead of 1. If the receiver of the message uses “majority” decoding, what is the probability that the message will be wrong when decoded? What independence assumptions are you making?

A man claims to have extrasensory perception. As a test, a fair coin is flipped 10 times and the man is asked to predict the outcome in advance. He gets 7 out of 10 correct. What is the probability that he would have done at least this well if he had no ESP?

In Exercises 19–22, find the area of the parallelogram whose vertices are listed. (0,-2),(5,-2),(-3,1),(2,1)

Let A and P be square matrices, with P invertible. Show that $\text{det}(PAP^{-1})=\text{det }A$

In Exercises 24–26, use determinants to decide if the set of vectors is linearly independent $$\begin{bmatrix} 4 \\\\ 6 \\\\ 2 \end{bmatrix}, \begin{bmatrix} -7 \\\\ 0 \\\\ 7 \end{bmatrix}, \begin{bmatrix} -3 \\\\ -5 \\\\ -2 \end{bmatrix}$$

In Exercises 21–23, use determinants to find out if the matrix is invertible. $$\begin{vmatrix} 5 & 1 & -1 \\\\ 1 & -3 & -2 \\\\ 0 & 5 & 3 \end{vmatrix}$$

Find the determinants in Exercises 5–10 by row reduction to echelon form. \begin{vmatrix} 1 & 3 & 2 & -4 \\\\ 0 & 1 & 2 & -5 \\\\ 2 & 7 & 6 & -3 \\\\ -3 & -10 & -7 & 2 \end{vmatrix}

In Exercises 39 and 40, A is an $n \times n$ matrix. Mark each statement True or False. Justify each answer. a. The cofactor expansion of det *A* down a column is equal to the cofactor expansion along a row. b. The determinant of a triangular matrix is the sum of the entries on the main diagonal.

Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants in Exercises 15–18.**Warning:** *This trick does not generalize in any reasonable way to $4\times4$ or larger matrices.* $$\begin{vmatrix} 0 & 3 & 1 \\\\ 4 & -5 & 0 \\\\ 3 & 4 & 1 \\\\ \end{vmatrix}$$

Compute the determinants in Exercises 9–14 by cofactor expansions. At each step, choose a row or column that involves the least amount of computation. \begin{vmatrix} 6 & 3 & 2 & 4 & 0 \\\\ 9 & 0 & -4 & 1 & 0 \\\\ 8 & -5 & 6 & 7 & 1 \\\\ 2 & 0 & 0 & 0 & 0 \\\\ 4 & 2 & 3 & 2 & 0 \\\\ \end{vmatrix}

Compute the determinants in Exercises 1–8 using a cofactor expansion across the first row. In Exercises 1–4, also compute the determinant by a cofactor expansion down the second column. $$\begin{vmatrix} 1 & 2 & 4 \\\\ 3 & 1 & 1 \\\\ 2 & 4 & 2 \\\\ \end{vmatrix}$$