Probability
1.
Three screws are drawn at random from a lot of 100 screws, 10 of which are defective. Find the probability of the event that all
3 screws drawn are non-defective, assuming that we draw
(a)
with replacement,
(b)
without replacement.
2.
In Problem 1 above, find the probability of obtaining at least 1 defective screw
(a)
directly,
(b)
by using complements.
3.
If a box contains 10 left-handed and 20 right-handed screws, what is the probability of obtaining at least one right-handed screw
in drawing 2 screws with replacement?
4.
If a certain kind of tire has a life exceeding 40,000 miles with probability 0.90, what is the probability that a set of these tires on
a car will last longer than 40,000 miles?
5.
If a circuit contains four automatic switches and we want that, with a probability of 99%, during a given time interval the switches
to be all working, what probability of failure per time interval can we admit for a single switch?
6.
A pressure control apparatus contains 3 electronic tubes. The apparatus will not work unless all tubes are operative. If the
probability of failure of each tube during some interval of time is 0.04, what is the corresponding probability of failure of the
apparatus?
Probability Distributions
1.
Graph the probability function f(x) = kx2 for x = 1,2,3,4,5 and constant k, and the frequency distribution.
2.
Graph the probability function f(x) = kx2 for 0 ≤ x ≤ 5 and constant k, and the frequency distribution.
3.
Graph f and F when the density of X is
( k = const, −2 ≤ x ≤ 2
f(x) =
0,
elsewhere.
Find P(0 ≤ X ≤ 2).
4.
A box contains 4 right-handed and 6 left-handed screws. Two screws are drawn at random without replacement. Let X be the
number of left-handed screws drawn. Find the probabilities
(a)
P(X = 0),
(b)
P(X = 1),
(c)
P(X = 2),
(d)
P(1 < X < 2),
(e)
P(X ≤ 1),
(f)
P(X ≥ 1),
(g)
P(X > 1), and
(h)
P(0.5 < X < 1)
5.
Let X be the number of years before a certain kind of pump needs replacement. Let X have the probability function
f(x) = kx3,
Find k. Sketch f and F.
x = 0,1,2,3,4.
6.
Let X [millimetres] be the thickness of washers. Assume that X has the density f(x) = kx if 0.9 < x < 1.1 and 0 otherwise. Find k.
What is the probability that a washer will have thickness between 0.95 mm and 1.05 mm?
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Q8.
Linear Regression Equations
Exercise
Use the following data to answer the questions given below.
i)
ii)
iii)
iv)
x
5
4
6
3
6
8
y
10
16
8
11
9
6
Calculate the coefficient of linear correlation and interpret your result.
Determine the regression equation.
Find y when x = 2.
Find y when x = 7.
FOURIER SERIES
Exercise
Solve the following differential equations:
FIRST ORDER ODINARY
DIFFERENTIAL EQUATIONS
Separable Integration Method
Exercise
Solve the following:
Integrating Factor Method
Exercise
Solve the following:
Bernoulli’s Equation Method
Exercise
Solve the following:
Homogeneous Differential Equation Method
Exercise
Solve the following:
Poisson Distribution
1. A company makes electric motors. The probability an electric motor is defective is 0.01. What is the probability
that a sample of 300 electric motors will contain exactly 5 defective motors?
Solution:
The average number of defectives in 300 motors is μ = 0.01 × 300 = 3
The probability of getting 5 defectives is:
2. Vehicles pass through a junction on a busy road at an average rate of 300 per hour.
a) Find the probability that none passes in a given minute.
b) What is the expected number passing in two minutes?
c) Find the probability that this expected number actually pass through in a given two-minute period.
Solution:
The average number of cars per minute is:
3. If electricity power failures occur according to a Poisson distribution with an average of 3 failures every twenty
weeks, calculate the probability that there will not be more than one failure during a particular week.
Solution:
The average number of failures per week is:
"Not more than one failure" means we need to include the probabilities for "0
failures" plus "1 failure".
4. Twenty sheets of aluminium alloy were examined for surface flaws. The frequency of the number of sheets with
a given number of flaws per sheet was as follows:
What is the probability of finding a sheet chosen at random which contains 3 or more surface flaws?
Solution:
The total number of flaws is given by:
(0 × 4) + (1 × 3) + (2 × 5) + (3 × 2) + (4 × 4) + (5 × 1) + (6 × 1) = 46
So the average number of flaws for the 20 sheets is given by:
The required probability is:
5. A life insurance salesman sells on the average 3 life insurance policies per week. Use Poisson's law to calculate
the probability that in a given week he will sell some policies.
Solution:
Here, μ = 3
"Some policies" means "1 or more policies". We can work this out by finding 1 minus the "zero policies"
probability:
P(X > 0) = 1 − P(x0)
So
6. Consider a computer system with Poisson job-arrival stream at an average of 2 per minute. Determine the
probability that in any one-minute interval there will be
(i) 0 jobs;
(ii) Exactly 2 jobs;
(iii) At most 3 arrivals.
Answers:
(i) 0.135
(ii) 0.2707
(iii) 0.8507 [Hint: p(0)+p(1)+p(2)+p(3)]
7. Suppose the average number of lions seen on a 1-day safari is 5. What is the probability that tourist will see less
than 4 lions on the next 1-day safari?
Answer: 0.2650
Binomial Distribution
1. A company owns 400 laptops. Each laptop has an 8% probability of not working. You randomly select 20
laptops for your sales people.
(a) What is the likelihood that 5 will not be working?
(b) What is the likelihood that they all will be working?
(c) What is the likelihood that they will be in a non-working condition?
Solution: Here n = 20, p = 8% = 0.08 and q =1-p = 1-0.08=0.92
(a) All 5 are not working. i.e. x=5.
(b) All are working( x=0)
(c) All are not working. (x=20)
2. A study indicates that 4% of American teenagers have tattoos. You randomly sample 30 teenagers. What is the
probability that exactly 3 will have a tattoo?
Solution:
Here n = 30, x=3 , p=4%=0.04 and q=1‐p=0.96
3. An XYZ cell phone is made from 55 components. Each component has a 0.002 probability of being defective.
What is the probability that an XYZ cell phone will not work properly?
Solution:
P(XYZ cell phone will work)(i.e. 0 defective components)
P(XYZ cell phone will not work properly) = 1‐0.896 = 0.1047
4. The ABC Company manufactures toy robots. About 1 toy robot per 100 does not work. You purchase 35 ABC
toy robots. What is the probability that exactly 4 do not work?
Solution: Here p = 1/100 = 0.01, n=35, x=4
5. The LMB Company manufactures tyres. They claim that only 0.007 of LMB tyres are defective. What is the
probability of finding exactly 2 defective tyres in a random sample of 50 LMB tyres?
Answer: 0.0428
6. An HDTV is made up of 100 components. Each component has a 0.005 probability of being defective. What is
the probability that an HDTV will not work perfectly? (Answer: 0.394)