probability

The average bit error rate at the input of a (7, 4, 1) Hamming decoder is 0.10. The probability that the decoder will fail to decode a received word correctly is ____. (rounded off...

A stick of length one meter is broken at two locations at distances of b₁ and b₂ from the origin (0), as shown in the figure. Note that 0 < b₁ < b₂ < 1. Which one of the following...

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different col...

A source transmits symbol S that takes values uniformly at random from the set {-2,0,2}. The receiver obtains Y = S + N, where N is a zero-mean Gaussian random variable independent...

The random variable X takes values in {-1,0,1} with probabilities P(X = -1) = P(X = 1) = α and P(X = 0) = 1 − 2α, where 0 < α < 1/2. Let g(α) denote the entropy of X (in bits), par...

Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable X denote the sum of the outcomes obtained. The expectation of X is ________ (rounded off...

X and Y are Bernoulli random variables taking values in {0,1}. The joint probability mass function of the random variables is given by: $P(X = 0, Y = 0) = 0.06$ $P(X = 0, Y = 1) =...

Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained. The expectation of $X$ is ___________ (roun...

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different col...

Suppose X and Y are independent and identically distributed random variables that are distributed uniformly in the interval [0,1]. The probability that X≥Y is _________.

Suppose $X$ and $Y$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability that $X \geq Y$ is _______...

Out of 1000 individuals in a town, 100 unidentified individuals are covid positive. Due to lack of adequate covid-testing kits, the health authorities of the town devised a strateg...

Consider communication over a memoryless binary symmetric channel using a (7, 4) Hamming code. Each transmitted bit is received correctly with probability (1 $$-$$ $$\in$$), and fl...

A box contains the following three coins. I. A fair coin head on one face and tail on the other face. II. A coin with heads to both the faces. III. A coin with tails on both the fa...

In a high school having equal number of boy students and girl students, 75% of the students study Science and the remaining 25% students study Commerce. Commerce students are two t...

Two continuous random variables X and Y are related as Y = 2X + 3. Let $$\sigma _x^2$$ and $$\sigma _y^2$$ denote the variances of X and Y, respectively. The variances are related...

The two sides of a fair coin are labelled as 0 and 1 . The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses....

A digital communication system transmits a block of $N$ bits. The probability of error in decoding a bit is $\alpha$. The error event of each bit is independent of the error events...

A random variable X takes values -1 and +1 with probabilities 0.2 and 0.8, respectively. It is transmitted across a channel which adds noise N, so that the random variable at the c...

Let X 1 , X 2 , X 3 and X 4 be independent normal random variables with zero mean and unit variance. The probability that X 4 is the smallest among the four is _______.

A cab was involved in a hit and run accident at night. You are given the following data about the cabs in the city and the accident. (i) 85% of cabs in the city are green and the r...

Three fair cubical dice are thrown simultaneously. The probability that all three dice have the same number of dots on the faces showing up is (up to third decimal place) _________...

Passengers try repeatedly to get a seat reservation in any train running between two stations until they are successful. If there is $$40$$% chance of getting reservation in any at...

Two random variables $$X$$ and $$Y$$ are distributed according to $$${f_{X,Y}}\left( {x,y} \right) = \left\{ {\matrix{ {\left( {x + y} \right),} & {0 \le x \le 1,} & {0 \le y \le 1...

The bit error probability of a memoryless binary symmetric channel is $${10^{ - 5}}$$. If $${10^{ - 5}}$$ bits are sent over this channel, then the probability that not more than o...

An analog baseband signal, band limited to 100 Hz, is sampled at the Nyquist rate. The samples are quantized into four message symbols that occur independently with probabilities $...

The probability of getting a ''head'' in a single toss of a biased coin is $$0.3.$$ The coin is tossed repeatedly till a ''head'' is obtained. If the tosses are independent, then t...

Let $$\,\,X \in \left\{ {0,1} \right\}\,\,$$ and $$\,\,Y \in \left\{ {0,1} \right\}\,\,$$ be two independent binary random variables. If $$\,\,P\left( {X\,\, = 0} \right) = p\,\,$$...

Ram and Ramesh appeared in an interview for two vacancies in the same department. The probability of Ram’s selection is 1/6 and that of Ramesh is 1/8. What is the probability that...

Let $$X \in \left\{ {0,1} \right\}$$ and $$Y \in \left\{ {0,1} \right\}$$ be two independent binary random variables. If $$P\left( {X\,\, = 0} \right)\,\, = p$$ and $$P\left( {Y\,\...

Let the random variable $$X$$ represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of $$X$$ is _______...

Ram and Ramesh appeared in an interview for two vacancies in the same department. The probability of Ram's selection is $$1/6$$ and that of Ramesh is $$1/8$$. What is the probabili...

A fair die with faces $$\left\{ {1,2,3,4,5,6} \right\}$$ is thrown repeatedly till $$'3'$$ is observed for the first time. Let $$X$$ denote the number of times the dice is thrown....

Suppose $$A$$ & $$B$$ are two independent events with probabilities $$P\left( A \right) \ne 0$$ and $$P\left( B \right) \ne 0.$$ Let $$\overrightarrow A $$ & $$\overrightarrow B $$...

Parcels from sender $$S$$ to receiver $$R$$ pass sequentially through two post - offices. Each post - office has a probability $${1 \over 5}$$ of losing an incoming parcel, indepen...

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution fun...

Let $$X$$ be a real - valued random variable with $$E\left[ X \right]$$ and $$E\left[ {{X^2}} \right]$$ denoting the mean values of $$X$$ and $${{X^2}}$$, respectively. The relatio...

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier of future calls, the probability distribution fun...

In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has...

A fair coin is tossed repeatedly till both head and tail appear at least once. The average number of tosses required is ________.

Let $$\,{X_1},\,\,{X_2}\,\,$$ and $$\,{X_3}\,$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,1} \right].$$ The probabili...

Let $$X$$ be a zero mean unit variance Gaussian random variable. $$E\left[ {\left| X \right|} \right]$$ is equal to ______

Let $${X_1},{X_{2,}}$$ and $${X_{3,}}$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,1} \right].$$ The probability $$P\l...

You are given three coins: one has heads on both faces, the second has tails on both faces, and the third has a head on one face and a tail on the other. You choose a coin at rando...

Let $${X_1},\,{X_2},$$ and $${X_3}$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,\,1} \right]$$. The probability $$P\le...

An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is

Let U and V be two independent zero mean Gaussian random variables of variances $${{1 \over 4}}$$ and $${{1 \over 9}}$$ respectively. The probability $$P(\,3V\, \ge \,\,2U)$$ is

Consider two identically distributed zero - mean random variables $$U$$ and $$V.$$ Let the cumulative distribution functions of $$U$$ and $$2V$$ be $$F(x)$$ and $$G(x)$$ respective...

Let $$U$$ and $$V$$ be two independent zero mean Gaussian random variables of variances $${1 \over 4}$$ and $${1 \over 9}$$ respectively. The probability $$\,P\left( {3V \ge 2U} \r...

What is the chance that a leap year, selected at random, will contain 53 Sundays?

A source alphabet consists of N symbols with the probability of the first two symbols being the same. A source encoder increases the probability of the first symbol by a small amou...

A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is

A and B are friends. They decide to meet between 1 PM and 2 PM on a given day. There is a condition that whoever arrives first will not wait for the other for more than 15 minutes....

Two independent random variable X and Y are uniformly distributed in the interval [ - 1, 1]. The probability that max [X, Y] is less than 1/2 is

Two independent random variables $$X$$ and $$Y$$ are uniformly distributed in the interval $$\left[ { - 1,1} \right].$$ The probability that max $$\left[ {X,Y} \right]$$ is less th...

A fair dice is tossed two times. The probability that the $$2$$ nd toss results in a value that is higher than the first toss is

A fair coin is tossed independently four times. The probability of the event ''The number of times heads show up is more than the number of times tails show up'' is

A fair coin is tossed $$10$$ times. What is the probability that only the first two tosses will yield heads?

A memory less source emits n symbols each with a probability p. The entropy of the source as a function of n

If $$E$$ denotes expectation, the variance of a random variable $$X$$ is given by

During transmission over a certain binary communication channel, bit errors occurs independently with probability p. The probability of at most one bit in error in a block of n bit...

If E denotes expectation, the variance of a random variable X is given by

An examination consists of two papers, paper $$1$$ and paper $$2.$$ The probability of failing in paper $$1$$ is $$0.3$$ and that in paper $$2$$ is $$0.2.$$ Given that a student ha...

A fair dice is rolled twice. The probability that an odd number will follow an even number is

The PDF of a Gaussian random variable X is given by $${p_x}(x) = \,{1 \over {3\sqrt {2\pi } }}\,\exp \,[ - \,{(x - 4)^2}/18]$$. The probability of the event {X = 4} is

During transmission over a communication channel, bit errors occur independently with probability 'p'. If a block of n bits is transmitted, the probability of at most one bit error...