Suppose $A$ is $n \times n$ and the equation $Ax=b$ has a solution for each $b$ in $\Bbb R^n$. Explain why $A$ must be invertible. [Hint: Is $A$ row equivalent to $I_n$]

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Explain why the columns of an $n \times n$ matrix $A$ span $\Bbb R^n$ when $A$ is invertible. [Hint: Review Theorem 4 in Section 1.4.]

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Suppose $A$, $B$, and $X$ are $n \times n$ matrices with $A$, $X$, and $A-AX$ invertible, and suppose
$(A-AX)^{-1}=X^{-1}B$ (eq. 3)
a. Explain why $B$ is invertible
b. Solve (eq. 3) for $X$. If you need to invert a matrix, explain why that matrix is invertible

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Suppose $A$ and $B$ are $n \times n$, B is invertible, and AB is invertible. Show that $A$ is invertible [Hint: Let $C=AB$, and solve this equation for $A$

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If $u$ and $v$ are in $\Bbb R^n$, how are $u^Tv$ and $v^Tu$ related? How are $uv^T$ and $vu^T$ related?

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Suppose $A$ is an $3 \times n$ matrix who columns span $\Bbb R^3$. Explain how to construct an $n \times 3$ matrix $D$ such that $AD=I_3$

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A and B are involved in a duel. The rules of the duel are that they are to pick up their guns and shoot at each other simultaneously. If one or both are hit, then the duel is over. If both shots miss, then they repeat the process. Suppose that the results of the shots are independent and that each shot of A will hit B with probability pA, and each shot of B will hit A with probability pB. What is
(a) the probability that A is not hit?
(b) the probability that both duelists are hit?
(c) the probability that the duel ends after the nth round of shots?
(d) the conditional...

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Barbara and Dianne go target shooting. Suppose that each of Barbara’s shots hits a wooden duck target with probability p1, while each shot of Dianne’s hits it with probability p2. Suppose that they shoot simultaneously at the same target. If the wooden duck is knocked over (indicating that it was hit), what is the probability that
(a) both shots hit the duck?
(b) Barbara’s shot hit the duck?
What independence assumptions have you made?

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Independent flips of a coin that lands on heads with probability p are made. What is the probability that the first four outcomes are
(a) H, H, H, H?
(b) T, H, H, H?
(c) What is the probability that the pattern T, H, H, H occurs before the pattern H, H, H, H?
Hint for part (c): How can the pattern H, H, H, H occur first?

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Prostate cancer is the most common type of cancer found in males. As an indicator of whether a male has prostate cancer, doctors often perform a test that measures the level of the prostate specific antigen (PSA) that is produced only by the prostate gland. Although PSA levels are indicative of cancer, the test is notoriously unreliable. Indeed, the probability that a noncancerous man will have an elevated PSA level is approximately .135, increasing to approximately .268 if the man does have cancer. If, on the basis of other factors, a physician is 70 percent certain that a
male has...

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There are 3 coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?

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Twelve percent of all U.S. households are in California. A total of 1.3 percent of all U.S. households earn more than \$250,000 per year, while a total of 3.3 percent of all California households earn more than \$250,000 per year.
(a) What proportion of all non-California households earn more than $250,000 per year?
(b) Given that a randomly chosen U.S. household earns more than $250,000 per year, what is the probability it is a California household?

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On rainy days, Joe is late to work with probability .3; on nonrainy days, he is late with probability .1. With probability .7, it will rain tomorrow.
(a) Find the probability that Joe is early tomorrow.
(b) Given that Joe was early, what is the conditional probability that it rained?

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Ms. Aquina has just had a biopsy on a possibly cancerous tumor. Not wanting to spoil a weekend family event, she does not want to hear any bad news in the next few days. But if she tells the doctor to call only if the news is good, then if the doctor does not call, Ms. Aquina can conclude that the news is bad. So, being a student of probability, Ms. Aquina instructs the doctor to flip a coin. If it comes up heads, the doctor is to call if the news is good and not call if the news is bad. If the coin comes up tails, the doctor is not to call. In this way, even if the doctor doesn’t call,...

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Suppose that 5 percent of men and .25 percent of women are color blind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females. What if the population consisted of twice as many males as females?

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In a certain community, 36 percent of the families own a dog and 22 percent of the families that own a dog also own a cat. In addition, 30 percent of the families own a cat. What is
(a) the probability that a randomly selected family owns both a dog and a cat?
(b) the conditional probability that a randomly selected family owns a dog given that it owns a cat?

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A certain town with a population of 100,000 has 3 newspapers: I, II, and III. The proportions of townspeople who read these papers are as follows:
I: 10 percent I and II: 8 percent I and II and III: 1 percent
II: 30 percent I and III: 2 percent
III: 5 percent II and III: 4 percent
(The list tells us, for instance, that 8000 people
read newspapers I and II.)
(a) Find the number of people who read only one
newspaper.
(b) How many people read at least two
newspapers?
(c) If I and III are morning papers and II is an
evening paper, how many people read at least
one...

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In an experiment, die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample space of this experiment? Let En denote the event that n rolls are necessary to complete the experiment. What points of the sample space are contained in En? What is $\big( \bigcup_1^\infty E_n \big)^c$

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A box contains 3 marbles: 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box and then replacing it in the box and drawing a second marble from the box. Describe the sample space. Repeat when the second marble is drawn without replacing the first marble.

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If 12 people are to be divided into 3 committees of respective sizes 3, 4, and 5, how many divisions are possible?

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