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# Probability

### Find the probability of selecting each of the following...

Each individual letter of the word "pfeffernuesse" is placed on a piece of paper, and all 13 pieces of paper are placed in a hat. Three letters are selected at random from the hat. Find the probability of selecting each of the following. a) with replacement b) without replacement 1) Pulling T

### Probability Randomly Selected Groups

A committee of four is to be randomly selected from a group of seven teachers and eight students. Find the probability that the committee will consist of four students.

### 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.

### Bernoulli Trials : The probability of a bit error in a memoryless binary symmetric communication channel is 10^-3. Find the probability of a block of 1000 bits has five or more errors.

The probability of a bit error in a memoryless binary symmetric communication channel is 10^-3. Find the probability of a block of 1000 bits has five or more errors.

### Probability : Percentage of Defective Chips and Normal Distribution

A student needs 12 chips of a certain type to build a circuit. It is known that 5% of these chips are defective. At least how many chips should she/he buy for there to be a greater than 95% probability of having enough chips for the circuit?

### 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 andfortx. (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}. (

### Probability : Selection Without Replacement and Stirling's Formula

A box contains 3n red balls and 3n white balls. 2n balls are selected at random without replacement, express the probability p that the balls selected contain n red and n white. Use Stirling's formula to obtain the approximation p ~ root(3/(2*pi*n))

### Given that the battery still works at 20, 000 km, what is the probability that it lasts for at least another 20, 000km? At 40,000km?

Suppose the probabilities that a new car battery functions For at least 20, 000km is 0.8 For at least 40, 000km is 0.5, For at least 60, 000km is 0.1. (i) Given that the battery still works at 20, 000 km, what is the probability that it lasts for at least another 20, 000km? (ii) Given that the battery still works at 20

### Probability and Statistics : Tree Diagram

Two friends Dave and Pete often play squash and tennis together. Over the years, they have found that Dave wins 3 out of every 5 rounds of squash and 1 out of every 4 tennis matches. If they play one match of squash and then one tennis match: a) Draw a tree diagram to describe the situation b) Find the probability that Dave

### Statistics and Probability

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

### Famous distribution that describes the frequency of the number of times a number comes up in a series of dice rolls

I cannot find the famous distribution that describes the frequency of the number of times a number comes up in a series of dice rolls? I think the answer is the distribution was discovered by Siméon-Denis Poisson (1781-1840). In probability theory and statistics, the Poisson distribution is a discrete probability distribut

### 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

### Probability of Scoring a Hole-In-One in Golf

See the attached files. Recall that the case study in your text concerns an amazing event that occurred during the second round of the 1989 US Open at Oak Hill in Pittsford, New York. Four golfers--Doug Weaver, Mark Wiebe, Jerry Pate, and Nick Price--made holes-in-one on the sixth hole. Recall from your text that the probabi

### Probability for Different Events

The sign "I LOVE MATHEMATICS" is put on the wall of the mathematics building at South Central Carolina Technical College. The letters start to fall off of the sign. Find the probability of each of the following events. 1.. The first letter that falls off is "M" . 2. The second letter that falls off is an "A" knowing tha

### 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

### Table Showing the PDF and CDF

5. Let E be the event that the number which comes up when a single die is tossed is divisible by 3. Let X be the number of times that event E occurs in 3 tosses of the die. (a) Make a table showing the pdf and the cdf of X. (b) Find E(X) and V(X).

### 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 of Cards Drawn

3. There are 13 standard decks of cards. Each deck is shuffled, separately. The 13 decks are lined up in a row on the table. The top card is drawn from each deck. What is the probability that at least 3 cards of each suit are drawn?

### 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 & Finance

Problem: The APR for a 30 year, \$250,000 mortgage at 6% interest compounded monthly and two discount points is: A. 6.25% b. 6.19% c. 6.12% d. 5.87% 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

### Probability & Statistics : Quartiles

The test scores of 30 students are listed Below. Find Q3; 31 41 45 48 52 55 56 56 63 65 67 67 69 70 70 74 75 78 79 79 80 81 83 85 85 87 90 92 95 99