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# Probability

### Combinations and Subsets : Probabilities

A committee consists of 8 married couples. In how many ways can a subcommitee of 5 people be chosen so that at most one married couple belongs to the subcommittee?

### Measures of Central Tendency : Mean, Median and Mode; Weighted Approach; Frequency of Numbers from a Pair of Dice

1. Each of the three measures of central tendency-the mean, the median, and the mode-are more appropriate for certain populations than others. For each type of measure, give two additional examples of populations where it would be the most appropriate indication of central tendency. 2. Find the mean, median, and mode of

### Process Flow Time - Queueing Problem and Critical Assumptions

See the attached file. Fort Lost-in-the-Woods is a basic training center for new Army recruits. Upon arriving at the military post the new recruits are processed through the Induction Center that involves three steps: background information gathering, medical examination, and barracks assignment. Arriving inductees first enter

### Basic Statistics , measures of central tendency

1. Each of the three measures of central tendency-the mean, the median, and the mode-are more appropriate for certain populations than others. For each type of measure, give two additional examples of populations where it would be the most appropriate indication of central tendency. 2. Find the mean, median, and mode of th

### Probability and Statistics: Varibables and Functions

A4. A4. Let X be a discrete random variable with range RX = {1, 2, ...}. (a) Show that ..... (b) Suppose that X has probability mass function (pmf) .... (i) Find the cumulative distribution function (cdf) of X. (ii) Use (a) to find E(X). IMPORTANT: Could you please add at the end of the question a list of formulae under

### Frequency Table

A secretary periodically checks to see how mnay of the three lines into the office are busy. Her findings for one week were the following: No. Lines busy Frequency 0 20 1 65 2 25 3

### Probability : Independent Dual Probability

A student applies for two different scholarships. The probability of receiving the first scholarship is 0.3 and the probability of receiving the second is 0.4. The decisions are made independently. Find the probability the student receives exactly one scholarship. A. 0.12 B. 0.42 C. 0.46 D. 0.40

### Probability : If the probability that at least one person makes an A on the final exam is 0.15, then the probability no one makes an A is...

If the probability that at least one person makes an A on the final exam is 0.15, then the probability no one makes an A is: A. 0.15 B. 0.65 C. 0.85 D. 0

### Probability : If a program is selected at random, the probability it is infected with a virus is...

A computer store has 10 copies of a word processing program, 12 copies of a spreadsheet program, and 8 copies of a draw program. Three of the word processing, four of the spreadsheet and two of the draw programs are infected with a computer virus. If a program is selected at random, the probability it is infected with a virus

### Sampling and Probability : The sample space of an experiment is {A, B, C, D} and P(A) = 0.1, P(B) = 0.3, P(C) = 0.4, P({A, C}) =

The sample space of an experiment is {A, B, C, D} and P(A) = 0.1, P(B) = 0.3, P(C) = 0.4, P({A, C}) = A. 0.4 B. 0.5 C. 0.6 D. 0.8

### 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 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

### 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 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 : Drawing Cards and Sampling Without Replacement

3 cards are drawn in succession from a regular straight deck of 52 playing cards. Find the probability that: (a) the first card is a Red Ace. (b) the second card is a 10 or Jack. (c) the third card is greater than 3 but less than 7.

### 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-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

### 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

### 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 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?

### Accounts Receivables Statistics - Tree Diagrams and Applications

An accountant found in a study that receivables fell into four categories: A: paid on time B: paid early C: paid late D: didn't pay Of a sample of 120 receivables she found that 35 were paid on time, 40 were paid early, 28 were paid late and the remainder were uncollectable. a) Using the results from the sample deter

### Queueing Theory : Swimmers and Exponential Distribution

Queueing Theory Question 1 An average of 10 people per hour arrive (inter-arrival times are exponential) intending to swim laps at the local YMCA. Each intends to swim an average of 30 minutes. The YMCA has 3 lanes open for lap swimming. If one swimmer is in a lane, he or she swims up and down the right side of the lane.