Probability theory
Probability: A numerical measure of the chance that an event will occur. Experiment: A process that generates well defined outcomes.
Sample space: The set of all experimental outcomes.
Sample point: An element of the sample space. A sample point represents an experimental outcome. Tree diagram: A graphical representation that helps in visualizing a multiple step experiment. Classical method: A method of assigning probabilities that is appropriate when all the experimental outcomes are equally likely. Relative frequency method: A method of assigning probabilities that is appropriate when data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated large number of times. Subjective method: A method of assigning probabilities on judgment. Event: A collection of sample points.

Complement of A : The event consisting of all sample points that are not in A. Venn diagram: A graphical representation for showing symbolically the sample space and operations involving events in which the sample space is represented by a rectangle and events are represented as circles within the sample space. Union of A and B : The event consisting of all sample points belonging to A or B or both. Intersection of A and B : The event containing the sample points belonging to both A and B. Conditional probabilities: the probability of an event given that another event already occurred. Joint probability: The probability of two events both is occurring that is the probability of the intersection of two events. Marginal probabilities: The values in the margins of a joint probability table that provides the probabilities of each event separately. Independent events: Two events A and B where the events have no influence on each other. Priory probabilities: Initial estimates of the probabilities of events. Posterior probabilities: revised probabilities of events based on additional information. Bayes’ theorem: A...

...
a. Suppose Evan chose a bottle from the refrigerator at random. Could we realistically say that the probability of choosing a diet soda is 7/3? Why or why not?
b. If there are 16 total bottles of diet soda, 8 total bottles of regular soda, and 4 total bottles of water, what is the probability of each of the following:
(i) Choosing a bottle of diet soda when a bottle is chosen at random
(ii) Choosing a bottle of regular soda when a bottle is chosen at random
(iii) Choosing a bottle of water when a bottle is chosen at random
Assessment 9-2B: Exercises 1 & 5
Suppose an experiment consists of spinning X and then spinning Y, as follows:
Find the following:
a. The sample space S for the experiment
b. The event A consisting of outcomes from spinning an even number followed by an even number
c. The event B consisting of outcomes from spinning at least one 2
d. The event C consisting of outcomes from spinning exactly one 2
Following are two boxes containing colored and white balls. A ball is drawn at random from box 1. Then a ball is drawn at random from box 2, and the colors of balls from both boxes are recorded in order.
Find each of the following:
a. The probability of two white balls
b. The probability of at least one colored ball
c. The probability of at most one colored ball
d. The probability of...

...I. ProbabilityTheory
* A branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.
* The word probability has several meanings in ordinary conversation. Two of these are particularly important for the development and applications of the mathematical theory of probability. One is the interpretation of probabilities as relative frequencies, for which simple games involving coins, cards, dice, and roulette wheels provide examples.
* It is the likeliness of an event happening based on all the possible outcomes. The ratio for the probability of an event 'P' occurring is P (event) = number of favorable outcomes divided by number of possible outcomes.
Example:
A coin is tossed on a standard 8×8 chessboard.
What is the theoretical probability that the coin lands on a black square?
Choices:
A. 0.5
B. 0.25
C. 0.42
D. 0.6
Correct answer: A
Solution:
Step 1: Theoretical probability = number of favorable outcomes / number of possible outcomes.
Step 2: The probability of the coin lands on the black square is 32.
Step 3: Total number of outcomes = 64.
Step 4: P (event) =
Step 5: == 0.5
Step 6: The theoretical...

...Worksheet 5 (Chapter 3): Probability II
Name: ______________________________________________
Section: _________________________
For any of the following questions be sure to show appropriate work and give appropriate probability
statements.
1. Students taking the Graduate Management Admissions Test (GMAT) were asked about their
undergraduate major and intent to pursue their MBA as a full-time or part-time student. A summary of
their responses follows.
Intended
Enrollment
Status
Full-Time
Part-Time
Totals
Undergraduate Major
Business
Engineering
352
197
150
161
502
358
Other
251
194
445
Totals
800
505
1305
a. If a student intends to attend classes full-time in pursuit of an MBA degree, what is the probability
that the student was an undergraduate engineering major?
P(Engineering | full-time) = 197/800 = 0.2463
b. If a student was an undergraduate business major, what is the probability that the student intends to
attend classes full-time in pursuit of an MBA degree?
P(full time | business) = 352/502 = 0.7012
c. Let A denote the event that the student intends to attend classes full-time in pursuit of an MBA
degree, and let B denote the event that the student was an undergraduate business major. Are events
A and B independent? Justify your answer.
Can use either method
P(A) = 800/1305 = 0.6130
P(A) = 800/1305 = 0.6130
P(A|B) = 352/502 = 0.7012
P(B) =...

...chapter, you will be able to ONEDefine probability. TWO Describe the classical, empirical, and subjective approaches to probability. THREEUnderstand the terms experiment, event, outcome, permutation, and combination. FOURDefine the terms conditional probability and joint probability. FIVE Calculate probabilities applying the rules of addition and multiplication. SIXUse a tree diagram to organize and computeprobabilities. SEVEN Calculate a probability using Bayes theorem. What is probability There is really no answer to this question. Some people think of it as limiting frequency. That is, to say that the probability of getting heads when a coin is tossed means that, if the coin is tossed many times, it is likely to come down heads about half the time. But if you toss a coin 1000 times, you are not likely to get exactly 500 heads. You wouldnt be surprised to get only 495. But what about 450, or 100 Some people would say that you can work out probability by physical arguments, like the one we used for a fair coin. But this argument doesnt work in all cases, and it doesnt explain what probability means. Some people say it is subjective. You say that the probability of heads in a coin toss is 12 because you have no reason for thinking either heads or tails more likely you might change your view if you knew...

...what is the probability of selecting:
A: 3 females and 2 males?
B: 4 females and 1 male?
C: 5 females?
D: At least 4 females?
Problem 2
By examining the past driving records of drivers in a certain city, an insurance company has determined the following (empirical) probabilities:
[pic]
If a driver in this city is selected at random, what is the probability that:
A: He or she drives less than 10,000 miles per year or has an accident? (Type a decimal)
B: He or she drives 10,000 or more miles per year and has no accidents? (type a decimal)
Problem 3
In a study to determine frequency and dependency of color-blindness relative to females and males, 1000 people were chosen at random and the following results were recorded:
[pic]
A: Convert the table to a probability table by dividing each entry by 1,000.
[pic]
B: What is the probability that a person is a woman, given that the person is color-blind? (Round to the nearest thousandth if needed)
C: What is the probability that a person is color-blind, given that the person is male?
D: Are the events color-blindness and male independent?
E: Are the events color-blindness and female dependent?
Problem 4
After careful testing and analysis, an oil company is considering drilling in two different sites. It is estimated that site A will net $40 million if successful (probability .4) and...

...Exponential Distribution
• Deﬁnition: Exponential distribution with parameter
λ:
λe−λx x ≥ 0
f (x) =
0
x s).
=
=
=
=
=
P (X > s + t|X > t)
P (X > s + t, X > t)
P (X > t)
P (X > s + t)
P (X > t)
e−λ(s+t)
e−λt
e−λs
P (X > s)
– Example: Suppose that the amount of time one
spends in a bank is exponentially distributed with
mean 10 minutes, λ = 1/10. What is the probability that a customer will spend more than 15
minutes in the bank? What is theprobability
that a customer will spend more than 15 minutes in the bank given that he is still in the bank
after 10 minutes?
Solution:
P (X > 15) = e−15λ = e−3/2 = 0.22
P (X > 15|X > 10) = P (X > 5) = e−1/2 = 0.604
2
– Failure rate (hazard rate) function r(t)
r(t) =
f (t)
1 − F (t)
– P (X ∈ (t, t + dt)|X > t) = r(t)dt.
– For exponential distribution: r(t) = λ, t > 0.
– Failure rate function uniquely determines F (t):
F (t) = 1 − e
3
t
− 0 r(t)dt
.
2. If Xi, i = 1, 2, ..., n, are iid exponential RVs with
mean 1/λ, the pdf of n Xi is:
i=1
(λt)n−1
,
fX1+X2+···+Xn (t) = λe−λt
(n − 1)!
gamma distribution with parameters n and λ.
3. If X1 and X2 are independent exponential RVs
with mean 1/λ1, 1/λ2,
λ1
.
P (X1 < X2) =
λ1 + λ2
4. If Xi, i = 1, 2, ..., n, are independent exponential
RVs with rate µi. Let Z = min(X1, ..., Xn) and
Y = max(X1, ..., Xn). Find distribution of Z and
Y.
– Z is an exponential RV with rate n µi.
i=1
P (Z > x)...

...Karan negi
12.2
12.3 We use equation 2 to find out probability:
F(t)=1 – e^-Lt
1-e^-(0.4167)(10) = 0.98 almost certainty. This shows that probability of another arrival in the next 10 minutes.
Now we figure out how many customers actually arrive within those 10 minutes. If the mean is 0.4167, then
0.4167*10=4.2, and we can round that to 4.
X-axis represents minutes (0-10)
Y-axis represents number of people.
We can conclude from this chart that the highest point with the most visitors is in the beginning of the 10 minutes. There may be a dispersion of visitors between the times, which according to this would be the slowest times. We can see 1 customer also visiting at the end of the 10 minutes. If a curve was to be drawn on this graph, it would signify a decline in visitors from point 0 to 6 and a steady move from 6 to 10.
16.3
The Lower- Colorado River Authority (LCRA) has been studying congestion at the boat-launching ramp near Mansfield Dam. On weekends, the arrival rate averages 5 boaters per hour, Poisson distributed. The average time to launch or retrieve a boat is 10 minutes, with negative exponential distribution. Assume that only one boat can be launched or retrieved at a time.
a.) The LCRA plans to add another ramp when the average turnaround time exceeds 90 minutes. At what average arrival rate per hour should the LCRA begin to consider adding another ramp?
b.) If there were room to park only two boats at the...

...Probability and its Applications
Published in association with the Applied Probability Trust
Editors: S. Asmussen, J. Gani, P. Jagers, T.G. Kurtz
Probability and its Applications
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Bass: Diffusions and Elliptic Operators. 1997
Bass: Probabilistic Techniques in Analysis. 1995
Berglund/Gentz: Noise-Induced Phenomena in Slow-Fast Dynamical Systems:
A Sample-Paths Approach. 2006
Biagini/Hu/Øksendal/Zhang: Stochastic Calculus for Fractional Brownian Motion
and Applications. 2008
Chen: Eigenvalues, Inequalities and Ergodic Theory. 2005
Costa/Fragoso/Marques: Discrete-Time Markov Jump Linear Systems. 2005
Daley/Vere-Jones: An Introduction to the Theory of Point Processes I: Elementary
Theory and Methods. 2nd ed. 2003, corr. 2nd printing 2005
Daley/Vere-Jones: An Introduction to the Theory of Point Processes II: General
Theory and Structure. 2nd ed. 2008
de la Peña/Gine: Decoupling: From Dependence to Independence, Randomly
Stopped Processes, U-Statistics and Processes, Martingales and Beyond. 1999
de la Peña/Lai/Shao: Self-Normalized Processes. 2009
Del Moral: Feynman-Kac Formulae. Genealogical and Interacting Particle
Systems with Applications. 2004
Durrett: Probability Models for DNA Sequence Evolution. 2002, 2nd ed. 2008
Ethier: The...