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Probability

Probability: Rayleigh Distributions

A random variable y has a Rayleigh distribution, if and only if its probability distribution is given by: f(y)=2(alpha)y^(-alpha y^2) , y>0 and alpha>0 f(y)=0 , elsewhere a) Show that mean=(1/2)*squareroot(pi/alpha) b) Show that the variance =(1/alpha)*(1-pi/4)

Probability and Standard Deviation

A Recent survey indicates that the average salary of all entry-level women managers in St. Paul is $56,700 with a standard deviation $7,200. What is the probability that a random sample of 50 such women will yield a mean entry-level that exceeds $58,000?

Probability Possibilities Associated with Weapons

The single shot probability of kill of any weapon system (gun, missle or slingshot) is less than 1 due to the reliability factors if no other reason. Suppose the single shot probability of kill of a new defensive missle system is 0.75%, and the the probability is not considered adequate. One familiar strategy to increase the o

The ELISA test was introduced in the mid-1980s to screen blood

The ELISA test was introduced in the mid-1980s to screen blood for the presence of AIDS antibodies. When antibodies are present ELISA is positive 98% of the time; when the blood lacks the antibidies the ELISA is positive 7% of the time. Assuming that 1% of the population has AIDS antibodiesin their blood, what is the probabili

Probability : Expected Values and Probability Distributions

According to an automobile manufacturer, the company uses 3,000 lock-and-key combinations on it's vehicles. Suppose that you find a key for one of those cars. a) What is the expected number of vehicles that you would have to check to find one that fits your key? b) What is the probability that you would have to check

Modelling the Rules of a Game

Alex and Mark are playing a game. The goal is to get to 100. The first player picks a whole number from 1 to 10, inclusive, and then the second player picks a whole number from 1 to 10 and adds it to the score so far. The first player repeats this move. They continue this way. The player who makes the score exactly 100 wins.

Probability / Statistics : Combination Application Word Problems

In a marketing survey, consumers are asked to give their first three choices, of 9 different drinks. In how many different ways can they indicate their choices? Find the present value of an ordinary annuity with annual payments of $1,000, for 6 years, at 10% interest compounded annually. A class consists of 15 students.

Probability, Sets and Counting

All of the students at a college are majoring in psychology, business, or both. 73% of the students are psychology majors & 62% are business majors. If there are 200 students, how many of them are majoring in both psychology & business?

Probability and Expected Value.

The numbers 1 through 9 are written individually on nine cards. Choose three cards from the nine, letting x, y, and z denote the numbers of the cards arranged in increasing order. A. There are ____ ____ such as x, y, and z combinations. B. The probability of having x, y, and z all even is . C. The probability of ha

Probability Mass Function, Mean and Variance

A lot consisting of 100 fuses is inspected by the following procedure: 5 fuses are selected randomly, and if all 5 "blow" at the specified amperage, the lot is accepted. Suppose that the lot contains 10 defective fuses. Find the probabily of accepting the lot. HINT: Let X ba a r.v. equil to the number of defective fuses in

Random Variables : Probability, Mean and Variance

Let the continuous r.v.X denote the weight (in pounds) of a package. The range of weight of the package is between 45 and 60 pounds. (a) Determine the probability that a package weighs more than 50 pounds. (b) Find the mean and the variance of the weight of packages. HINT: Assume that X is uniformly distributed over (45

Random Variables : Mean and Variance and Error Probability

Binary data are transmitted over a noisy communications channel in block of 16 binary digits. The probability that a received digit is in error as a result of channel noise is 0.01. Assume that the errors occurring in various digit positions within a block are independent. (a) Find the mean and the variance of the number of

Probability-Conditional Prob. and Independence

The Question is: Suppose that each time that you buy a car, you choose between Ford and General Motors. Suppose that each time after the first, you stay with the same company with probability 2/3 and switch with probability 1/3. If you are equally likely to choose either company for your first car, what is the probability that

Probability-Bayes' Theorem

The Question is: Suppose that 400 pregnant women take a home pregnancy test, and 397 of them test "positive" and the other 3 test "negative." Suppose also that 200 nonpregnant women take the test, and 184 of them test "negative" and the remaining 16 test "positive." What is the probability that a woman who tests positive is ac

Probability - Bayes' Theorem

The Question is: An estimated 8% of men and O.5% of women are colorblind. If a colorblind person is selected at random, what is the probability that the person is a man? (Assume that men and women occur in equal numbers) Now if we have to solve using Bayes' Theorem, how are we supposed to set the problem up and represent the

Probability to find a partner

The Question is: In the game of bridge, each of 4 players is dealt 13 cards. If a certain player has no aces, find the probability that that person's partner has: a.)no aces b.)at least 2

Probability: Multiple Choice Test Example

In class we are learning about conditional probability and independence. The question is: On a multiple-choice test you know the answers to 70% of the question (and get them right), and for the remaining 30% you choose randomly among the 5 answers. What percent of the answers should you expect to get right?

Bayes Theorem: Conditional Probability

In answering a question on a multiple choice test, a student either knows the answer or he guesses. Let p be the probability that he knows the answer and 1-p the probability that he guesses. Assume that a student who guesses at the answer will be correct with probability 1/m, where m is the number of multiple choice alternativ

Minimal Cut Sets and Probability of System Life

Consider a structure in which the minimal path sets are {1, 2, 3} and {3, 4, 5} a. What are the minimal cut sets? b. If the component lifetimes are independent uniform (0,1) random variables, determine the probability that the system life will be less than ½ .

Mutually exclusive events

Please see attached file. --- Let A, B be mutually exclusive events and P(A) = 0.4, P(B) = 0.3 Find... ---

Probability - Frequency Of Dice Rolls

4. When a pair of dice is rolled, the total will range from 2 (1,1) to 12 (6,6). It is a fact that some numbers will occur more frequently than others as the dice are rolled over and over. A. Why will some numbers come up more frequently than others? B. Each die has six sides numbered from 1 to 6. How many possible ways can

Probability: Coin flips

Six people will decide which of them are on a committee by flipping a coin. Each person flips the coin, and is on the committee if he or she gets a head. What is the probability that someone is on the committee, but not all six people?

Probability : Distribution and Expected Value

A large sports conference moved to have women compose at least 40% of its athletics within 5 years. Suppose they exactly achieve the 40% figure, and that 5 athletics are picked at random from 10 universities. The number of women is recorded. (a) give a probability distribution (b) find the expected value

Probability and Independent Events : Bayes Theorem

5. (Sudden death) The NHL has another season-long strike, but the owners and players reach an agreement in June which leaves time for a highly abbreviated season. They decide that fans want to see the Stanley Cup decided, and so they play only a sudden-death version of the seventh game of the final round of the playoffs. Her

Probability : Independent Events and Dice Rolls

1. Three dice, 1, 2, and 3, are rolled independently. ? Event A12 is that dice 1 and 2 show the same number. ? Event A13 is that dice 1 and 3 show the same number. ? Event A23 is that dice 2 and 3 show the same number. (a) Are events A12 and A13 independent? (b) Are the three events independent?