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Probability

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 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

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.

Cumulative Distribution Function, Renewal Function and Poisson

3. Let {N(t)}>0 be a renewal process for which the interarrival times {T1,T2,. . .} have cumulative distribution function F. Recall that the renewal function m(t) is given by m(t) E[N(t)j. (a) Find E[N(t)Ti=x] fort<x andfortx. (b) Find E[N(t)] E[E[N(t) T1]] to prove that m(t) satisfies the renewal equation m(t) F(t) + f m(t

Stochastic Processes : Poisson Process and Markov Chains

1. Suppose that shocks occur according to a Poisson process with rate A> 0. Also suppose that each shock independently causes the system to fail with probability 0 < p < 1. Let N denote the number of shocks that it takes for the system to fail and let T denote the time of the failure. (a) FindP{T>tNrrn}. (b) FindP{NrrnT=t}. (

Statistics : Standard Deviation and Mean (10 Problems)

1. True or False? The standard deviation of a population will always be smaller than the standard deviation of the sample means (of the samples used to estimate this same population). 2. The mean height for a population is 65 inches and the standard deviation is 3 inches. Let X_ denote the mean height for a sample of people

Statistics : Probability of Winning a Powerball Lottery

The Powerball For a single ticket, a player first selects five numbers from the numbers 1-53 and then chooses a powerball number, which can be any number between 1 and 42. A ticket costs $ 1. In the drawing, five white balls are drawn randomly from 53 white balls numbered 1-53, and one red Powerball is drawn randomly from

18 Problems : Payoff Tables and Decision Trees; Control Charts; Bayes Theorem; Total Quality Management ( Demings 14 Points of Management ), Expected Monetary Value, Red Bead Experiment

1. A tabular presentation that shows the outcome for each decision alternative under the various states of nature is called a: a. payback period matrix. b. decision matrix. c. decision tree. d. payoff table. 2. The difference between expected payoff under certainty and expected value of the best act without certainty is

Concepts of probability statistics

1. Event A: You spend your entire Memorial Day 2006 in Acapulco. Event B: You spend your entire Memorial Day 2006 with some of your friends. TRUE OR FALSE: A and B are mutually exclusive events. 2.Event A: You spend your entire Memorial Day 2004 in Denver. Event B: You spend your entire Memorial Day 2004 in Vermont. TRU

Independent random variables

1. A coin is tossed 3 times. Discrete random variable X is equal to the number of times Heads comes up. Discrete random variable Y has the value 1 if the first toss comes up heads and 0 otherwise. (a) Find Pr[(X=1)n(y=1)] Are X and Y independent random variables?

Probability - drawing tokens

7. There are some tokens in a bag. Each token has a number ( a positive integer) printed on it. Ernesto does not know how many tokens are in the bag, and the only thing he knows about the numbers on them is that the mean of the numbers is 3 and the variance is 2. Ernesto is going to play a game in which he draws 1 token from th

Probability and Cumulative Distribution Functions

4. A hand of 5 cards contains 2 red cards and 3 black cards. Trish plays the following game: A card is drawn from the hand. If the card is red, the game stops immediately. If the card is black, this black card is set aside and a red card is put into the hand in its place. Then another card is drawn from the hand and the same pro

Probability

1. There are 3 boxes. Each box contains several envelopes. Some envelopes have "you lose" written on them. The rest say "you win..." . However, of the ones that say "you win..." , some contain only a piece of paper saying "... nothing" . Each of the other contains a $5 bill. The first box contains 3 "lose" envelopes and 2 so-

Probabilities

Problem 4. Let X denote the number of boys in a family with four children. Pr(X > 3) is? a. 5/16 b. ¼ c. 11/16 d. 2/3 e. None of the above

Probability & Statistics : Chebychev Inequality

Suppose that a probability distribution has a mean 20 and standard deviation 3. The Chebychev inequality states that the probability that an outcome lies between 16 and 24 is: a. less than 7/16 b. at least 7/16 c. at most 1/4 d. at least 1/4 e. none of the above

Probability & Statistics : Outcome of Dice Rolls

Two people play a game. A single die is thrown. If the outcome is a 2 or a 3, then player A pays player B $6.00. How much should B pay A when a 1, 4, 5, or 6 is thrown so that A and B break even, on average, over many repititions of the game? a. $2.00 b. $8.00 c. $4.00 d. $6.00 e. None of the above

Sets, Counting and Probability : Events

A light Bulb manufacturer tests a light bulb by letting it burn until it burns out. The experiment consists of observing how long (in hours) the light bulb burns. Let E be the event "the bulb lasts less than 100 hours", F be the event "the bulb lasts less than 50 hours",and G be the event "the bulb lasts more than 120 hours." T

Frequencies of Dice Roll Combinations and Probabilities of Dice Rolls

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 a nu

Probability : Maximum Likelihood Estimator and p-Values

A GMAT test center reports that out of 200 students who took a GMAT test at the test center 60 scored above 600. a) Derive the maximum likelihood estimator (MLE) of p, the proportion of students scoring above 600. What is the ML estimate of n*p(1-p) b) What is the MLE of P(X<50)? c) Test if the proportion of students sc

Probability distribution

A single die is rolled once. You lose $12 if a number divisible by 3 turns up. How much should you win if a number not divisible by 3 turns up for the game to be fair? How would I set this up, and what formulas?

Probability : Sampling Without Replacement - Card Hands

6. A hand of 4 cards contains one card of each suit ( I,e., one Heart, one Diamond, One Club and one spade). Simon uses this hand to play a game for which the rules are as follows: (1) When a Heart is drawn, the game stops immediately. (2) If the Diamond is drawn, the Diamond is set aside and not replaced (so that when the game

Combinations and Probability (8 Problems) - Camp Cawapati

1. At Camp Cawapati, there is a prize ceremony at the end of each camp session. The campers compete at a variety of challenges throughout the 2-week session, in hopes of earning trophies. For each challenge a camper wins, the camper receives a small replica of the Sacred Cawapati Cup. (These replica trophies are all identical.)

Probabilities related to a sample mean.

The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi. a. What is the probability that the sample mean breaking strength for a random sample of 40 rivets is between 9900 and 10,200. b. If the sample size had been 15 rather than 40, could the probability requested in part (a) b

Joint Probability Distribution Function

10. Annie and Alvie have agreed to meet between 5:00 pm and 6:00 pm for dinner at a local health food restaurant. Let X=Annie's arrival time and Y=Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6]. a. What is the joint pdf of X and Y? b. What is the probability that