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If mathematics is the queen of sciences, then probability is the queen of applied mathematics. The concept of probability originated in the seventeenth century and can be traced to games of chance and gambling. Games of chance include actions like drawing a card, tossing a coin, selecting people at random and noting number of females, number of calls on a telephone, frequency of accidents, and position of a particle under diffusion. Today, probability theory is a wellestablished branch of mathematics that finds applications from weather predictions to share market investments. Mathematical models for random phenomena are studied using probability theory.
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Determine the sample space for each of the following random experiments.
A student is selected at random from a probability and statistics lecture class, and the student’s total marks are determined.
A coin is tossed three times, and the sequence of heads and tails is observed.
One urn contains three red balls, two white balls, and one blue ball. A second urn contains one red ball, two white balls, and three blue balls:
One ball is selected at random from each urn. Describe the sample space.
If the balls in two urns are mixed in a single urn and then a sample of three is drawn, find the probability that all three colors are represented when sampling is drawn (i) with replacement (ii) without replacement.
A fair coin is continuously flipped. What is the probability that the first five flips are (i) H, T, H, T, T (ii) T, H, H, T, H.
The first generation of a particle is the number of offsprings of a given particle. The next generation is formed by the offsprings of these members. If the probability that a particle has
k offsprings (split into
k parts) is
\(p_k\) where
\(p_0 =0.4,p_1=0.3,p_2=0.3\), find the probability that there is no particle in the second generation. Assume that the particles act independently and identically irrespective of the generation.
A fair die is tossed once. Let
A be the event that face 1, 3, or 5 comes up,
B be the event that it is 2, 4, or 6, and
C be the event that it is 1 or 6. Show that
A and
C are independent. Find
P(
A,
B, or
C occurs).
An urn contains four tickets marked with numbers 112, 121, 211, 222, and one ticket is drawn at random. Let
\(A_i ~ (i = 1, 2, 3)\) be the event that
ith digit of the number of the ticket drawn is 1. Discuss the independence of the events
\(A_1, A_2\) and
\(A_3\).
There are two identical boxes containing, respectively, four white and three red balls; three white and seven red balls. A box is chosen at random, and a ball is drawn from it. Find the probability that the ball is white. If the ball is white, what is the probability that it is from the first box?
Let
A and
B are two independent events. Show that
\(A^c\) and
\(B^c\) are also independent events.
Five percent of patients suffering from a certain disease are selected to undergo a new treatment that is believed to increase the recovery rate from 30 to 50%. A person is randomly selected from these patients after the completion of the treatment and is found to have recovered. What is the probability that the patient received the new treatment?
Four records lead away from the country jail. A prisoner has escaped from the jail. The probability of escaping is 1 / 6, if road 2 selected, the probability of success is 1 / 6, if road 3 is selected, the probability of escaping is 1 / 4, and if road 4 is selected, the probability of escaping is 9 / 10.
What is the probability that the prisoner will succeed in escaping?
If the prisoner succeeds, what is the probability that the prisoner escaped by using road 4 and by using road 1?
The probability that an airplane accident which is due to structure failure is identified correctly is 0.85, and the probability that an airplane accident which is not due to structure failure is identified as due to structure failure is 0.15. If 30% of all airplane accidents are due to structure failure, find the probability that an airplane accident is due to structure failure given that it has been identified to be caused by structure failure.
The numbers
\(1, 2, 3, \ldots , n\) are arranged in random order. Find the probability that the digits
\(1, 2, \ldots , k ~~(k < n)\) appear as neighbors in that order.
In a town of
\((n+1)\) inhabitants, a person tells a rumor to a second person, who in turn, repeats it to a third person, etc. At each step, the recipient of the rumor is chosen at random from the
n people available. Find the probability that the rumor will be told
r times without returning to the originator.
A secretary has to send
n letters. She writes addresses on
n envelopes and absentmindedly places letters one in each envelope. Find the probability that at least one letter reaches the correct destination.
A pond contains red and golden fish. There are 3000 red and 7000 golden fish, of which 200 and 500, respectively, are tagged. Find the probability that a random sample of 100 red and 200 golden fish will show 15 and 20 tagged fish, respectively.
A coin is tossed four times. Let
X denote the number of times a head is followed immediately by a tail. Find the distribution, mean, and variance of
X.
In a bombing attack, there is 50% chance that a bomb can strike the target. Two hits are required to destroy the target completely. How many bombs must be dropped to give a 99completely destroying the target?
For what values of
\(\alpha \) and
p does the following function represent a PMF
\( p_X(x) = \alpha p^x, ~~ x = 0,1,2,\ldots . \)
Let the probability density function of
X be given by
What is the value of
c?
What is the distribution of
X?
\(P \left( \frac{1}{2}< X < \frac{3}{2} \right) \)?
A bombing plane flies directly above a railroad track. Assume that if a larger(small) bomb falls within 40(15) feet of the track, the track will be sufficiently damaged so that traffic will be disrupted. Let
X denote the perpendicular distance from the track that a bomb falls. Assume that
Find the probability that a larger bomb will disrupt traffic.
If the plane can carry three large(eight small) bombs and uses all three(eight), what is the probability that traffic will be disrupted?
A random variable
X has the following PMF
Determine the value of
k.
Find
\(P(X<4), \ P(X \ge 5), \ P(0<X<4)\).
Find the CDF of
X.
Find the smallest value of
x for which
\(P(X\le x)=1/2\).
An urn contains
n cards numbered
\(1,2,\ldots ,n\). Let
X be the least number on the card obtained when
m cards are drawn without replacement from the urn. Find the probability distribution of random variable
X. Compute
\(P(X \ge 3/2)\).
Let
X be binomial distributed with
\(n=25\) and
\(p=0.2\). Find expectation, variance, and
\(P(X < E(X) 2 \sqrt{Var(X)})\).
Let
X be a Poisson distributed random variable such that
\(P[X=0]=0.5\). Find the mean of
X.
In a uniform distribution, the mean and variance are given by 0.5 and
\(\frac{25}{12}\), respectively. Find the interval on which the probability is uniform distributed.
Let
Is
\(F_{X}\) a distribution function? What type of random variable is
X? Find the PMF/PDF of
X?
Let
X be a continuous type random variable with PDF
If
\(E(X)=\frac{3}{5}\), find the value of
a and
b.
Let
X be a random variable with mean
\(\mu \) and variance
\(\sigma ^2\). Show that
\( E[ (aXb)^2]\), as a function of
b, is minimized when
\(b = \mu \).
Let
X and
Y be two random variables such that their MGFs exist. Then, prove the following:
If
\(M_X(t)=M_Y(t),~\forall t\), then
X and
Y have same distribution.
If
\(\varPsi _X(t)=\varPsi _Y(t),~\forall t\), then
X and
Y have same distribution.
Let
\(\varOmega =[0,1]\). Define
\(X:\varOmega \rightarrow \mathbb {R}\) by
For any interval
\(I \subseteq [0,1]\), let
\(P(I)=\displaystyle \int _I2xdx\). Determine the distribution function of
X and use this to find
\(P(X>1/2), \ P(1/4<X<1/2), \ P(X<1/2 / X>1/4)\).
A random number is chosen from the interval [0, 1] by a random mechanism. What is the probability that (i) its first decimal will be 3 (ii) its second decimal will be 3 (iii) its first two decimal will be 3’s?
Prove that, the random variable
X has exponential distribution and satisfies a memoryless property or Markov property which is given as
Suppose that diameters of a shaft s manufactured by a certain machine are normal random variables with mean 10 and s.d. 0.1. If for a given application the shaft must meet the requirement that its diameter falls between 9.9 and 10.2 cm. What proportion of shafts made by this machine will meet the requirement?
A machine automatically packs a chemical fertilizer in polythene packets. It is observed that
\(10\%\) of the packets weigh less than 2.42 kg while
\(15\%\) of the packets weigh more than 2.50 kg. Assuming that the weight of the packet is normal distributed, find the mean and variance of the packet.
Show that the PGF’s of the geometric, negative binomial and Poisson distribution exists and hence calculate them.
Verify that the normal distribution, geometric distribution, and Poisson distribution have reproductive property, but the uniform distribution and exponential distributions do not.
Let
\(Y\sim N(\mu ,\sigma ^2)\) where
\(\mu \in \mathbb {R}\) and
\(\sigma ^2 <\infty \). Let
X be another random variable such that
\(X=e^Y.\) Find the distribution function of
X. Also, verify that
\(E(\log (X))=\mu \) and
\(Var(\log (X))=\sigma ^2\).
Let
\(X \sim \) B(
n,
p). Use the CLT to find
n such that:
\(P[X > n/2] \le 1 \alpha \). Calculate the value of
n when
\(\alpha = 0.90\) and
\(p =0.45\).
Suppose that the number of customers who visit SBI, IIT Delhi on a Saturday is a random variable with
\(\mu = 75\) and
\(\sigma = 5\). Find the lower bound for the probability that there will be more than 50 but fewer than 100 customers in the bank?
Does the random variable
X exist for which
Suppose that the life length of an item is exponentially distributed with parameter 0.5. Assume that ten such items are installed successively so that the
ith item is installed immediately after the
\((i1)\)th item has failed. Let
\(T_i\) be the time to failure of the
ith item
\(i = 1,2,\ldots ,10\) and is always measured from the time of installation. Let
S denote the total time of functioning of the 10 items. Assuming that
\(T_i's\) are independent, evaluate
\(P(S \ge 15.5)\).
A certain industrial process yields a large number of steel cylinders whose lengths are distributed normal with mean 3.25 inches and standard deviation 0.05 inches. If two such cylinders are chosen at random and placed end to end what is the probability that their combined length is less than 6.60 inches?
A complex system is made of 100 components functioning independently. The probability that any one component will fail during the period of operation is equal to 0.10. For the entire system to function at least 85 of the components must be working. Compute the approximate probability of this.
Suppose that
\(X_i, i = 1, 2, \ldots , 450\) are independent random variables, each having a distribution
N(0, 1). Evaluate
\(P(X^2_1 + X^2_2 + \cdots + X^2_{450} > 495)\) approximately.
Suppose that
\(X_i, i = 1, 2, \ldots , 20\) are independent random variables, each having a geometric distribution with parameter 0.8. Let
\(S=X_1+\cdots +X_{20}\). Use the central limit theorem
\(P(X\ge 18)\).
A computer is adding number, rounds each number off to the nearest integer. Suppose that all rounding errors are independent and uniform distributed over (−0.5, 0.5).
(a) If 1500 numbers are added, what is the probability that the magnitude of the total error exceeds 15?
(b) How many numbers may be added together so that the magnitude of the total error is less than 10 with probability 0.90?
Let
\(X \sim B(n, p)\). Use CLT to find
n such that
\( P[ X > n/2 ] \ge 1  \alpha .\) Calculate the value of
n, when
\(\alpha = 0.90\) and
\(p = 0.45\).
A box contains a collection of IBM cards corresponding to the workers from some branch of industry. Of the workers,
\(20\%\) are minors and
\(30\%\) adults. We select an IBM card in a random way and mark the age given on this card. Before choosing the next card, we return the first one to the box. We observe
n cards in this manner. What value should
n have so that the probability that the frequency of cards corresponding to minors lies between 0.10 and 0.22 is 0.95?.
Items are produced in such a manner that the probability of an item being defective is
p (assume unknown). A large number of items say
n are classified as defective or nondefective. How large should
n be so that we may be
\(99\%\) sure that the relative frequency of defective differs from
p by less than 0.05?
A person puts some rupee coins into a piggybank each day. The number of coins added on any given day is equally likely to be 1, 2, 3, 4, 5 or 6 and is independent from day to day. Find an approximate probability that it takes at least 80 days to collect 300 rupees?
Suppose that 30 electronic devices say
\(D_1,D_2, \ldots ,D_{30}\) are used in the following manner. As soon as
\(D_1\) fails,
\(D_2\) becomes operative. When
\(D_2\) fails,
\(D_3\) becomes operative, etc. Assume that the time to failure of
\(D_i\) is an exponentially distributed random variable with parameter
\(=\) 0.1 (h)
\(^{1}\). Let
T be the total time of operation of the 30 devices. What is the probability that
T exceeds 350 h?
Suppose that
\(X_i\;,\;i=1,2,\ldots , 30\) are independent random variables each having a Poisson distribution with parameter 0.01. Let
\(S = X_1+X_2+ \cdots +X_{30}\).
(a) Using central limit theorem evaluate
\(P(S \ge 3)\).
(b) Compare the answer in (a) with exact value of this probability.
Use CLT to show that
Consider polling of
n voters and record the fraction
\(S_n\) of those polled who are in favor of a particular candidate. If
p is the fraction of the entire voter population that supports this candidate, then
\(S_n = \frac{X_1 + X_2 + \cdots + X_n}{n}\), where
\(X_i\) are independent Bernoulli distributed random variables with parameter
p. How many voters should be sampled so that we wish our estimate
\(S_n\) to be within 0.01 of
p with probability at least 0.95?
Let
\(X_1, X_2, \ldots \) be a sequence of independent and identically distributed random variables with mean 1 and variance 1600, and assume that these variables are nonnegative. Let
\(Y = \displaystyle \sum _{k=1}^{100} X_k\). Use the central limit theorem to approximate the probability
\(P(Y \ge 900)\).
If you wish to estimate the proportion of engineers and scientists who have studied probability theory and you wish your estimate to be correct within
\(2\%\) with probability 0.95, how large a sample should you take when you feel confident that the true proportion is less than 0.2?
1.
A student is selected at random from a probability and statistics lecture class, and the student’s total marks are determined.
2.
A coin is tossed three times, and the sequence of heads and tails is observed.
1.
One ball is selected at random from each urn. Describe the sample space.
2.
If the balls in two urns are mixed in a single urn and then a sample of three is drawn, find the probability that all three colors are represented when sampling is drawn (i) with replacement (ii) without replacement.
1.
What is the probability that the prisoner will succeed in escaping?
2.
If the prisoner succeeds, what is the probability that the prisoner escaped by using road 4 and by using road 1?
$$ f(x)=\left\{ \begin{array}{ll} c(4x2x^2), &{} 0< x < 2 \\ 0, &{} \text{ otherwise }\end{array}\right. .$$
1.
What is the value of
c?
2.
What is the distribution of
X?
3.
\(P \left( \frac{1}{2}< X < \frac{3}{2} \right) \)?
$$ f_{X}(x)=\left\{ \begin{array}{ll} \frac{100x}{5000}, &{} if\ x \in 0< x < 100 \\ 0, &{} \text{ otherwise }\end{array}\right. .$$
1.
Find the probability that a larger bomb will disrupt traffic.
2.
If the plane can carry three large(eight small) bombs and uses all three(eight), what is the probability that traffic will be disrupted?
$$ \begin{array}{cccccccccc} \hline X=x &{} 0 &{} 1 &{} 2 &{} 3 &{} 4 &{} 5 &{} 6 &{} 7 &{} 8\\ \hline P(X=x)&{} k &{} 3k &{} 5k &{} 7k &{} 9k &{} 11k &{} 13k &{} 15k &{} 17k \\ \hline \end{array} $$
1.
Determine the value of
k.
2.
Find
\(P(X<4), \ P(X \ge 5), \ P(0<X<4)\).
3.
Find the CDF of
X.
4.
Find the smallest value of
x for which
\(P(X\le x)=1/2\).
$$\begin{aligned} F_{X}(x) = \left\{ \begin{array}{ll} 0, &{} x < 0 \\ 1  2 e^{x} + e^{2 x}, &{} x \ge 0 \end{array} \right. . \end{aligned}$$
$$\begin{aligned} f_{X}(x) = \left\{ \begin{array}{ll} a+bx^2, &{} 0< x < 1 \\ 0 &{} ~~~ \text{ otherwise } \end{array} \right. \end{aligned}$$
1.
If
\(M_X(t)=M_Y(t),~\forall t\), then
X and
Y have same distribution.
2.
If
\(\varPsi _X(t)=\varPsi _Y(t),~\forall t\), then
X and
Y have same distribution.
$$ X(w)=\left\{ \begin{array}{ll} w, &{} 0 \le w \le 1/2\\ w1/2, &{} 1/2 \le w \le 1 \end{array}\right. . $$
$$\begin{aligned} P(X>x+s / X>s)=P(X>x) ~~ x, s \in \mathbb {R^{+}}. \end{aligned}$$
(2.21)
$$ P\left[ \mu  2 \sigma \le X \le \mu + 2 \sigma \right] = 0.6.$$
$$\begin{aligned} \lim _{n\rightarrow \infty }e^{nt}\displaystyle \sum _{k=0}^{n}\frac{(nt)^k}{k!}=1 = \left\{ \begin{array}{ll} 1, &{}0<t<1 \\ 0.5, &{} t=1\\ 0, &{} t>1 \end{array} \right. . \end{aligned}$$
1
2
3
Andrey Nikolaevich Kolmogorov (1903–1987) was a twentiethcentury Russian mathematician who made significant contributions to the mathematics of probability theory. It was Kolmogorov who axiomatized probability in his fundamental work, Foundations of the Theory of Probability (Berlin), in 1933.
Thomas Bayes (1702–1761) was a British mathematician known for having formulated a special case of Bayes’ Theorem. Bayes’ Theorem (also known as Bayes’ rule or Bayes’ law) is a result in probability theory, which relates the conditional and marginal probability of events. Bayes’ theorem tells how to update or revise beliefs in light of new evidence: a posteriori.
Johann Carl Friedrich Gauss (30 April 1777–23 February 1855) was a German mathematician who contributed significantly to many fields, including number theory, algebra, statistics. Sometimes referred to as “greatest mathematician since antiquity,” Gauss had exceptional influence in many fields of mathematics and science and is ranked as one of history’s most influential mathematicians. He discovered the normal distribution in 1809 as a way to rationalize the method of least squares.
Zurück zum Zitat Castaneda LB, Arunachalam V, Dharmaraja S (2012) Introduction to probability and stochastic processes with applications. Wiley, New York CrossRef Castaneda LB, Arunachalam V, Dharmaraja S (2012) Introduction to probability and stochastic processes with applications. Wiley, New York
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Zurück zum Zitat Feller W (1968) An introduction to probability theory and its applications: volume I. Wiley, New York MATH Feller W (1968) An introduction to probability theory and its applications: volume I. Wiley, New York
MATH
Zurück zum Zitat Rohatgi VK, Md. Ehsanes Saleh AK (2015) An introduction to probability and statistics. Wiley, New York CrossRef Rohatgi VK, Md. Ehsanes Saleh AK (2015) An introduction to probability and statistics. Wiley, New York
CrossRef
Zurück zum Zitat Ross SM (1998) A first course in probability. Prentice Hall, Englewood Cliffs Ross SM (1998) A first course in probability. Prentice Hall, Englewood Cliffs
 Titel
 Review of Probability
 DOI
 https://doi.org/10.1007/9789811317361_2
 Autoren:

Dharmaraja Selvamuthu
Dipayan Das
 Verlag
 Springer Singapore
 Sequenznummer
 2
 Kapitelnummer
 Chapter 2