Hey guys, this is the probability Assignment. Last date for submission is 10 aug...

Q1. What is the probability of picking a card that was either red or black?

Q2. A problem in statistics is given to 5 students A, B, C, D, E. Their chances of solving it are ½,1/3,1/4,1/5,1/6. What is the probability that the problem will be solved?

Q3. A person is known to hit the target in 3 out of 4 shots whereas another person is known to hit the target in 2 out of 3 shots. Find the probability that the target being hit at all when they both try?

Q4. An investment consultant predicts that the odds against price of a certain stock will go up during the next week are 2:1 and the odds in the favor of the price remaining the same are 1:3.What is the probability that the price of the stock will go down during eth next week?

Q5. A bag contains 10 White and 6 Black balls. 4 balls are successfully drawn out and not replaced. What is the probability that they are alternately of different colors?

Q6.In a multiple-choice question there are 4 alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks all the correct answers. The candidate decides to tick answers at random. If he is allowed up to 3 chances to answer the question, find the probability that he will get marks in the question?

Q7. A and B are two independent events. The probability that both occur simultaneously is 1/6 and the probability that neither occurs is 1/3. Find the probability of occurrence of event A and B separately?

Q8. Three screws are drawn at random from a lot of 10 screws containing 4 defective. Find the probability that all the 3 screws drawn are non-defective. Assuming the screws are drawn (i) With replacement (ii) Without replacement

Q9. What is the probability that a leap year selected at random will contain 53 Sundays?

Q10. An article manufactured by a company consists of two parts A and B. In the process of...

...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) =...

...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)...

...ProbabilityTheory and Game of Chance
Jingjing Xu
April 24, 2012
I. INTRODUCTION
Probabilitytheory is the mathematical foundation of statistics, and it can be applied to many areas requiring large data analysis. Curiously, that the study on probabilitytheory has its root in parlor games and gambling. In 17th century, dice gambling was a very common entertainment among the upper class. An Italian mathematician and gambler Gerolamo Cardano founded the concept of probability by studying the rules of rolling dice: since a die is a cube with each of its six faces showing a different number from 1 to 6, when it is rolled, the probability of seeing each number is equal. Therefore, some of the gamblers began to wonder, that taking a pair of dice and rolling them a couple of times, which has the larger probability of seeing a sum of 9 or seeing a sum of 10? What about seeing double sixes? In a correspondence between Blaise Pascal and Pierre Fermat, the problems were resolved, and this triggered the first theorem in the modern theory of probability.
II. BASIC DEFINITIONS
Definition 1
In probabilitytheory, the sample space,...

...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
Azencott et al.: Series of Irregular Observations. Forecasting and Model
Building. 1986
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...

...Problem Sheet-III
1. If X is uniformly distributed over (0, 10), calculate the probability that
a. X < 3 (Ans: 3/10)
b. X > 6 (Ans: 4/10)
c. 3 < X < 8. (Ans: 5/10)
2. Buses arrive at a specified stop at 15-minute intervals starting at 7 AM. That is, they arrive at 7, 7:15, 7:30, 7:45, and so on. If a passenger arrives at the stop at a time that is uniformly distributed between 7 and 7:30, find theprobability that he waits
d. Less than 5 minutes for a bus
e. More than 10 minutes for a bus.
3. A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 with a variance of 260. If a tax of 20% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 20% more expensive), what will be the variance of the annual cost of maintaining and repairing a car?
(Ans: 374)
4. The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without failing for at least five hours. (Ans: 0.42)
5. A company has two electric generators. The time until failure for each generator follows an exponential distribution with mean 10. The company will begin using the second generator immediately after the first one fails. What is the variance of...

...What is the mode of inheritance?
What is the mode of inheritance?
Probability: Probability is used to determine the chance of an outcome occurring in any one trial. It is equal to the expected proportion of an outcome in a series of events. Example: Outcome: X-bearing or Y-bearing sperm Events: All sperm in an ejaculate Trial: Fertilization of a single egg Probability: 1/2 for X, 1/2 for Y
• Law of Independence: Applies if the occurrence of an outcome in one trial does not influence the probability of another outcome in a subsequent trial. Example: Given that a couple has had one boy, the probability that their next offspring is male is still 1/2.
• Multiplication Rule: The combined probability of two or more independent outcomes happening in two or more trials is the product of their individual probabilities. [a and b - multiply] Example: The probability of a couple having two boys in row is 1/2 x 1/2 = 1/4.
• Addition Rule: The probability of two or more alternative outcomes happening in the same trial is the sum of their individual probabilities. [a or b - add] Example: The probability of a couple having either or boy or a girl is 1/2 + 1/2 = 1.
Waardenburg syndrome is an autosomal dominant condition that accounts for 1.4 percent of congenitally deaf persons. Given that a woman has Waardenburg...