probability

Consider discrete random variables X and Y with probabilities as follows: P(X = 0 and Y = 0) = $\frac{1}{4}$ P(X = 1 and Y = 0) = $\frac{1}{8}$ P(X = 0 and Y = 1) = $\frac{1}{2}$ P...

Let X and Y be continuous random variables with probability density functions Px(x) and Py(y), respectively. Further, let Y = X^2 and Px(x) = {1, x ∈ (0,1] {0, otherwise Which one...

Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y=X^2$ and $P_X(x)=\left\{\begin{array}{cc}1, &...

Given a fair six-faced dice where the faces are labelled '1', '2', '3', '4', '5', and '6', what is the probability of getting a '1' on the first roll of the dice and a '4' on the s...

The expected number of trials for first occurrence of a "head" in a biased coin is known to be 4. The probability of first occurrence of a "head" in the second trial is __________....

Given a fair six-faced dice where the faces are labelled '1', '2', '3', '4', '5', and '6', what is the probability of getting a '1' on the first roll of the dice and a '4' on the s...

The expected number of trials for first occurrence of a "head" in a biased coin is known to be 4. The probability of first occurrence of a "head" in the second trial is __________...

There are two identical dice with a single letter on each of the faces. The following six letters : Q, R, S, T, U and V, one on each of the faces. Any of the six outcomes are equal...

Let $X$ is a continuous random variable denoting the temperature measured. The range of temperature is $[0,100]$ degree Celsius and let the probability density function of $X$ be $...

Suppose the probability that a coin toss shows "head" is $p$, where $0

The probability of a resistor being defective is 0.02. There are 50 such resistors in a circuit. The probability of two or more defective resistors in the circuit (round off to two...

An urn contains 5 red balls and 5 black balls. In the first draw, one ball is picked at random and discarded without noticing its colour. The probability to get a red ball in the s...

An e-mail password must contain three characters. The password has to contain one numeral from 0 to 9, one upper case and one lower case character from the English alphabet. How ma...

A class of twelve children has two more boys than girls. A group of three children are randomly picked from this class to accompany the teacher on a field trip. What is the probabi...

Assume that in a traffic junction, the cycle of the traffic signal lights is 2 minutes of green (vehicle does not stop) and 3 minutes of red (vehicle stops). Consider that the arri...

A person decides to toss a fair coin repeatedly until he gets a head. He will make at most 3 tosses. Let the random variable Y denote the number of heads. The value of var{Y}, wher...

The probability that a k-digit number does NOT contain the digits 0, 5, or 9 is

There are 3 red socks, 4 green socks and 3 blue socks. You choose 2 socks. The probability that they are of the same colour is

A class of twelve children has two more boys than girls. A group of three children are randomly picked from this class to accompany the teacher on a field trip. What is the probabi...

A person decides to toss a fair coin repeatedly until he gets a head. He will make at most $$3$$ tosses. Let the random variable $$Y$$ denotes the number of heads. The value of var...

An urn contains $$5$$ red balls and $$5$$ black balls. In the first draw, one ball is picked at random and discarded without noticing its colour. The probability to get a red ball...

Assume that in a traffic junction, the cycle of the traffic signal lights is $$2$$ minutes of green (vehicle does not stop) and $$3$$ minutes of red (vehicle stops). Consider that...

Candidates were asked to come to an interview with $$3$$ pens each. Black, blue, green and red were the permitted pen colours that the candidate could bring. The probability that a...

Two coins $$R$$ and $$S$$ are tossed. The $$4$$ joint events $$\,\,\,\,\,\,{H_R}{H_S},\,\,\,\,{T_R}{T_S},\,\,\,\,{H_R}{T_S},\,\,\,\,{T_R}{H_S}\,\,\,\,\,\,\,$$ have probabilities $$...

Given Set $$\,\,\,A = \left\{ {2,3,4,5} \right\}\,\,\,$$ and Set $$\,\,\,B = \left\{ {11,12,13,14,15} \right\},\,\,\,$$ two numbers are randomly selected, one from each set. What i...

The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c$$ respectively. Of these subjects, the student has $$75$$% chance of passing in a...

Two players, $$A$$ and $$B,$$ alternately keep rolling a fair dice. The person to get a six first wins the game. Given that player $$A$$ starts the game, the probability that $$A$$...

Consider a die with the property that the probability of a face with $$'n'$$ dots showing up is proportional to $$'n'.$$ The probability of the face with three dots showing up is _...

A fair coin is tossed $$n$$ times. The probability that the difference between the number of heads and tails is $$(n-3)$$ is

A continuous random variable $$X$$ has a probability density function $$f\left( x \right) = {e^{ - x}},0 < x < \infty .$$ Then $$P\left\{ {X > 1} \right\}$$ is

A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, 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 box contains $$4$$ white balls and $$3$$ red balls. In succession, two balls are randomly selected and removed from the box. Given that first removed ball is white, the probabili...

Assume for simplicity that $$N$$ people, all born in April (a month of $$30$$ days) are collected in a room, consider the event of at least two people in the room being born on the...

$$X$$ is uniformly distributed random variable that take values between $$0$$ and $$1.$$ The value of $$E\left( {{X^3}} \right)$$ will be

A fair coin is tossed $$3$$ times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is

If $$P$$ and $$Q$$ are two random events, then which of the following is true?