1. Suppose that the monthly worldwide average of airplane crashes of commercial planes is 2.0. (a) What is the probability of exactly 6 crashes in the first 3 months? (b) What is the probability of having exactly 2 crashes in each of the first two months? (c) What is the probability of having exactly 2 crashes in the each of
1. Suppose that the number of eggs laid on a tree leaf by a particular type of insect is a Poisson random variable with l = 1. (a) What is the probability that any particular leaf will have at least two eggs? (b) Suppose that a person searches through leaves on a tree until he finds one with at least two eggs. Letting X
On a particular examination paper, a pupil whose mean score is {see attachment} will actually score X, where X has a {see attachment} distribution. Over the pupils taking the test, {see attachment} is assumed to have a {see attachment} distribution. Find the density of the test score of a randomly chosen pupil. Given that Fred
Assume that the proportion of commercial vehicles among users of the Humber Bridge varies randomly from day to day, with density {see attachment} over 0<x<1, where c is a constant. Show that c = 12, find the distribution function, and sketch the density and distribution functions over -1<x<2. On what fraction of days is the pro
Please specify the terms you use (if necessary) and explain each step of your solutions. In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the
Let X be such that P{X=1}= p = 1 - P{X = -1} Find c≠1 such that E...(See attachment for full question)
If X has distribution function F, what is the distribution function of the random variable aX + B, where a and B are constants, and a is not equal to zero. *(Please see attachment for complete question)
Suppose that a batch of 100 items contains 6 that are defective and 94 that are nondefective. If X is the number of defective items in a randomly drawn sample of 10 items from the batch, find (a) P{X=0} and (b) P{X>2} (Answer first if sampling with replacing, and then if sampling without replacing)
1) 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 0.6, independant of the outcomes of the other games. Find the probability that the stronger team wins the series in exactly i gam
Suppose that it takes at least 9 votes from a 12-member jury to convict a defendant. Suppose the probability that a juror votes a guilty person innocent is 0.2. whereas the probability that the juror votes an innocent person guilty is 0.1. If each juror acts independently and if 65% of the defendants are guilty, find the probabi
1. Suppose that two teams play a series of games that ends when one of the teams has one i number of games. Suppose that each game played is, independently, won by team A with probability p. Find the expected number of games that are played when (a) i = 2 and when (b) i = 3. Show also in both cases that this number is maximized
In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly he wins the amount (in dollars) equal to the sum of the fingers shown by him and by his opponent. If both players guess correctly or neither guesses
3.3. Assume that within a given service game at tennis, successive points form Bernoulli trials with p = P(Server wins) > ½. Tennis rules say that the service game ends as soon as either player has won at least four points, and is at least two points ahead of the other. Find the chances the server wins the game 4-0, 4-1, and 4-
Suppose that the height, in inches, of a 25-year-old man is a normal random variable with parameters μ = 71 and o^2 = 6.25. What percentage of 25-year-old men are over 6 feet 2 inches tall? What percentage of men in the 6-footer club are over 6 foot 5 inches? *(Please see attachment for proper symbols)
On a 12 by 20 rectangular board, three plane figures are drawn: a square 6 on a side , a circle with radius 4 and an equilateral triangle that is 8 on a side. If a dart is thrown that does hit the large rectangle what is the probabillity that it hits inside one of the three plane figures? (Landing on the side of a figure coun
3.2 A majority verdict of 10-2 or better may be permitted in a jury trial. Assuming each juror has probability 0.9 of reaching a Guilty verdict, and decides independently, what is the probability the jury decides to convict? (Partial solution provided in attachment) 3.14 For both the Gamma and Beta distribution, find the va
Show that, as in the binomial distribution, the successive probabilities in a Poisson distribution increase to a maximum, then decrease towards zero. Under what circumstance is there a unique maximum probability? Three partial solutions are attached.
Please see attachment for question. Please do these two things when solving the problem: i) Use words to describe the solution process ii) If you use a theorem or proposition, please state what it is called and your source. For example, THEOREM 3.1, ROSS, CHP 3.
2.17 Show that, if (a) a fair die is thrown times independently, it is more likely than not that at least one six appears; (b) a pair of fair dice are thrown 24 times independently, it is more likely than not that a double six does not appear. (This pair of calculation has an honoured place in the history of the developmen
For a different medical application of Bayes' theorem, suppose one person in 1000 suffers an adverse reaction to a drug, and a simple test for this reaction is on offer. The test is said to be 95% reliable, meaning that if the person would suffer a reaction, a positive result comes up 95% of the time, and if they would not have
Three balls are to be randomly selected with replacement from an urn containing 20 balls numbered 1 through 20. If we bet that at least one of the drawn balls has a number as large as or larger than 17, what is the probability that we win the bet?
The problems are from 400 level probability class but introductory course. Please specify the terms you use (if necessary) and explain each step of your solutions. If there is anything unclear in the problem, please tell me. Thank you very much.
20. A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is, in fact, guilty, each judge will independently vote guilty with probability 0.7, whereas when the defendant is, in fact, innocent, this probability drops to 0.2. If 70 percent of defendants are
The probability that a family with 6 children has exactly four boys is: a. 1/3 b. 1/64 c. 15/64 d. 3/8 e. none of the above
The problems are from 400 level probability class but introductory course. Please specify the terms you use (if necessary) and explain each step of your solutions. If there is anything unclear in the problem, please tell me. Thank you very much.
18. A true-false question is to be posed to a husband and wife team on a quiz show. Both the husband and the wife will, independently, give the correct answer with probability p. which of the following is a better strategy for the couple? (a) Choose one of them and let that person answer the question; or (b) have them both con
12. Urn I contains 2 white and 4 red balls, whereas urn II contains 1 white and 1 red ball. A ball is randomly chosen form urn I and put into urn II, and a ball is then randomly selected from urn II. What is (a) the probability that the ball selected from urn II is white; (b) the conditional probability that the transferred b
10. Ninety-eight percent of all babies survive delivery. However, 15 percent of all births involve Cesarean (C) sections, and when a C section is performed the baby survives 96 percent of the time. If a randomly chosen pregnant woman does not have a C section, what is the probability that her baby survives?
An ectopic pregnancy is twice as likely to develop when the pregnant woman is a smoker as it is when she is a nonsmoker. If 32 percent of women of childbearing age are smokers. What percentage of women having ectopic pregnancies are smokers?
6. If there are 12 strangers in a room, what is the probability that no two of them celebrate their birthday in the same month?