Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all 10! possible rankings are equally likely. Let X denote the highest rank-ing achieved by a woman. (For instance, X = 1 if the top-ranked person is female.) Find P{X = i}, i = 1, 2, 3, . . . , 8, 9, 10.

Show that the equivalence amount of citric acid for 1.00 g of so Part A: Equivalence Amounts Cup A Cup B 1. Mass of the cup, water and the stirrer 2. Mass of sodium bicarbonate 3. Mass of citric acid 4. Total mass (Add 1, 2, and 3) 5. Mass of the cup, the solution, and the stirrer after the reaction 6. Difference (Subtract 5 from 4) = Mass of carbon dioxide

Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win \$2 for each black ball selected and we lose \$1 for each white ball selected. Let X denote our winnings. What are the possible values of X, and what are the probabilities associated with each value?

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 $I_n$]

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.]

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

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 $A$

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?

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 $AD=I_3$

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...

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?

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?

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...

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?

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?

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?

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,...

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?