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Probability

Basic Probability

1. You just got that degree for which you have been working long and hard. You are offered a job, but the drawback is that you have to move. You check into the town and things start to look better. There are lots of nice parks, bike trails, community events, and a low crime rate. You were interested to find out that 2% of the pe

Probability : Addition and Multiplication Rules

It can be easy enough to get the addition rule and the multiplication rule confused. Tell me the difference between the two. Provide the notations and then tell me what type of problem I would use each one for. Addition Rule: Notation for Addition Rule: P(A or B) = P(event A occurs or event B occurs or they both occur). M

Jointly Distributed Random Variables : Binomial Distributions

Suppose that 25% of US males and 23% of US females never eat breakfast. Suppose a random sample of 250 men and 250 women is taken and polled about eating breakfast. a) What is the probabilty that at least 110 of these 500 people never eat breakfast? b) What is the probability that the number of women who never eat breakfast is

Probability : Mean and Variance of a Distribution

Show that, if X has a U(0,1) distribution, its mean is 1/2 and its variance is 1/12. Let Y = a + (b - a)X where a,b are constants, a<b. What is the distribution of Y? What are its mean and variance? Please see attached for partial solution.

Continuous Random Variables - Introductory Probability - 400 Level Class

30) An image is partitioned into 2 regions - one white and the other black. A reading taken from a randomly chosen point in the white section will give a reading that is normally distributed with Mean=4 and Variance =4, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading wi

Find the probability

Each item produced by a certain manufacturer is, independantly, of acceptable quality with probability 0.95. Approximate the probability that at most 10 of the next 150 items produced are unacceptable.

Odds

Find the odds in favor of a card drawn from an ordinary deck being the following. A red face card or a queen.

Form a related table with entries that are relative frequencies. Based on the relative frequencies, estimate the probability that an album selected at random is a rock album, classical CD or country western CD, jazz or classical album, CD.

A music store has jazz, classical, country western, and rock music albums on tapes and CDs in the following quantities: Jazz Classical Country Western Rock Total Tape 46 14 36 53 149 CD 38 53 22 19 132 Total 84 67 58 72 281 a. Form a related table with entries that are relative frequencies (three decimal places).

Probability : Density of the Sum of Independent Variables

We have found the density of the sum of two independent U(0,1) variables. Use that result to find the density of the sum of three independent U(0,1) variables. (You will need to treat intervals (0,1), (1,2) and (2,3) separately, although an appeal to symmetry can reduce the work. Give yourself a gold star if you get the righ

Probability (Pooled Test / Individual Tests)

Only answer parts C) and D) of the attached question, please. Each of 500 soldiers in an army company independently has a certain disease with probability 1/10[cubed]. This disease will show up in a blood test, and to facilitate matters blood samples from all 500 are pooled and tested. One of the 500 people is Jones, who kno

Important information about Basic Probability

Each of 500 soldiers in an army company independently has a certain disease with probability 1/10[cubed]. This disease will show up in a blood test, and to facilitate matters blood samples from all 500 are pooled and tested. a) What is the (approximate) probability that the blood test will be positive (and so at least one pe

Basic Probability (Averages)

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

Probability : Poisson Random Variables, Value and Variance

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

Probability : Mean, Distribution, Density and Exponential Distribution

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

Probability : Density and Distribution Functions - Sketch

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

Probability: Players Acting Independently

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

Probability :Distribution Function

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)

Probability : Ordered and Non-ordered Sets, Sampling and Replacing

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)

Introductory Probability

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

If each juror acts independently and if 65% of the defendants are guilty, find the probability that the jury renders a correct decision. What % of defendants are convicted?

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

Probability : Games Won

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

Probability : Bernoulli Trials and Exponential Distributions

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-

Probability : Probability Generating Function and Beta and Gamma Function

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

Probabilities in Poisson Distribution

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.

Probability

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.

Probability : General Probability, Systems and Dice

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

Probability : Bayes' Theorem and General Probability

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