Probability Questions

Two events $A$ and $B$ are independent and equally likely. If $P(A\cup B)=3/4$, then $P(A)=$ ?

In a box containing $100$ eggs, $10$ eggs are rotten. The probability that out of a sample of $5$ eggs none is rotten (if sampling is with replacement) is:

Let $A$ and $B$ be two mutually exclusive events. Let $P(\cdot )$ denote the probability. Which of the following statements is true?

The value of $C$ for which $P(x=K)=C{K}^2$ can serve as the probability function of a random variable $x$ that takes values $0,1,2,3,4$ is:

The probability density function for a continuous random variable $x$ is:$f(x)=\{\begin{array}{ll}\frac{x}{6},& 0\le x<2\\ \frac{1}{3},& 2\le x<4\\ -\frac{x}{6}+1,& 4\le x<6\end{array}$ The mean value of $x$ is:

If four dice are thrown together, what is the probability that the sum of the numbers appearing on them is 25?

$A$ and $B$ are two independent witnesses in a case. The probability that $A$ will speak the truth is $1/3$ and the probability that $B$ will speak the truth is $3/4$. If $A$ and $B$ agree on a certain statement, then the probability that the statement is true is:

For any two events $A$ and $B$, $P(A)=P(A|B)=1/4$ and $P(B|A)=1/2$. Then which of the following is NOT correct?

A fair die is thrown six times. The probability of getting a factor of 6 exactly four times is:

Below is a figure indicating a sample space $S$ and the probability of occurrences of events $A$ and $B$. Which colored section of the portion will indicate the probability of events $B$ and $A$ occurring simultaneously?

A continuous random variable $x$ has the probability density function:$f(x)=\{\begin{array}{ll}{e}^x,& -1\le x<0\\ \frac{1}{1+x^2},& 0\le x<1\\ \text{elsewhere},& e-1-\frac{\pi }{4}\end{array}$$P(-1<x<1)$ is equal to:

Suppose $P(A)=0.4$, $P(B)=P$, and $P(A\cup B)=0.7$. If $A$ and $B$ are independent events, then the value of $P$ is:

The chances of a defective screw in three boxes A, B, and C are $1/5$, $1/6$, and $1/7$ respectively. A box is selected at random and a screw is found to be defective. Find the probability that it came from box A.

If $P(A)=0.4$, $P(B)=0.8$, and $P(B|A)=0.6$, then $P(A\cup B)$ is equal to:

What is the probability of getting a numbered card when a card is drawn from a 52-card pack?

In the following table, $x$ is the discrete random variable and $p(x)$ is the probability density function. The standard deviation of $x$ is:| $x$ | $1$ | $2$ | $3$ | $4$ ||---|---|---|---|---|| $p(x)$ | $0.2$ | $0.4$ | $0.3$ | $0.1$ |

Given $P(H)=0.3$, $P(R)=0.4$, and $P(H|R)=0.2$. Find $P(R\cap H)$.

A fair die is tossed thrice. If the probabilities of zero, one, two, and three successes are $8/27,4/9,2/9,\text{ and }1/27$ respectively, find the mean of the number of successes.

Let $f(x)=c{x}^2$ for $x=1,2,3$. The value of constant $c$ such that $f$ satisfies the conditions of being a probability mass function (PMF) is:

In throwing a six-faced die, let A be the event that an even number occurs, B be the event that an odd number occurs, and C be the event that a number greater than 3 occurs. Which one of the following is correct?

A coin is tossed twice. If $E$ and $F$ denote the occurrence of a head on the first toss and second toss respectively, then what is $P(E\cup F)$ equal to?

Which of the following is true for binomial distribution?

Urn A consists of $3$ blue and $4$ green balls while another urn B consists of $5$ blue and $6$ green balls. One ball is drawn at random from one of the urns and it is found to be blue. Determine the probability that it was drawn from urn B?

Find the number of ways of picking $4$ cards from a set of $52$ playing cards if all four cards are of the same suit?

Consider the probability distribution function defined as $f_X(x)=\{\begin{array}{ll}|x|,& |x|<1\\ 0,& \text{otherwise}\end{array}$. Which of the following is true?

Let $X$ be a discrete random variable with probability $P(X=x)=\frac{1}{2^x}$ for $x=1,2,3,\dots$. Then the expected value of $X$ is:

If $X$ and $Y$ are random variables such that $E(X)=10$, $V(X)=25$ and $E(Y)=0$, $V(Y)=1$, the values of $a,b$ such that $Y=aX-b$ will be:

In a job placement event, there are $1000$ aspirants attending a company interview, out of which $430$ are females. It is known that $10\%$ of the females are graduates. Determine the probability that an aspirant selected randomly is a graduate given that the selected aspirant is a female?

For a continuous random variable $X$ distributed in the interval $[0,10]$, the probability $P(X=2)$ is:

The variance ${\sigma }^2$ of a random variable $X$ is given by:

A coin is tossed $5$ times. The probability of head is $\frac{1}{2}$. The probability of exactly $2$ heads is:

A card is drawn at random from a pack of $52$ cards. What is the probability that the card drawn is either an Ace or a King?

The mean of the probability distribution function given by the following table will be:\begin{center}\begin{tabular}{|c|c|c|c|c|c|}\hline$x$ & 1 & 2 & 3 & 4 & 5 \ \hline$P(x)$ & 0.2 & 0.35 & 0.25 & 0.15 & 0.05 \ \hline\end{tabular}\end{center}

Given that $x$ is a random variable in the range $[0,\infty )$ with a probability density function $f(x)=\frac{e^{-x/2}}{K}$, the value of the constant $K$ is:

$X$ is a random variable with probability density function:
$$f(x)=\{\begin{array}{ll}\frac{1}{2\pi }& -\pi \le x\le \pi \\ 0& \text{elsewhere}\end{array}$$
Then $E[\sin x]$ is:

Let $X$ be a random variable that is uniformly chosen from the set of positive odd numbers less than $100$. The expectation $E[X]$ is:

If the expected value of a random variable $X$ is 2 and its variance is 1, then what will be the variance of $3X+4$?

Five coins are tossed at a time. The probability of obtaining at least one head is:

What is the probability that the roots of the equation $x^2+x+n=0$ are real, where $n\in \mathbb{N}$ and $n<4$?

A coin is tossed $3$ times. If $X$ is a random variable representing the number of tails, then the mathematical expectation of $X$ is:

If the probability of $A$ to fail in an examination is $0.2$ and that for $B$ is $0.3$, then the probability that either $A$ or $B$ fails (meaning exactly one of them fails) is:

A family has two children. If the first child is a boy, find the probability that the other child is also a boy.

A player tosses three coins. If the first toss is a Head, find the probability that there are two consecutive Heads among the three coins.

If an event $E$ has only one sample point of a sample space, it is called a ______:

A die is thrown 3 times. Let Event A be "4 on the 3rd throw" and Event B be "6 on the 1st and 5 on the 2nd throw". What is the probability of A given that B has already occurred?

Let $X$ and $Y$ be two independent Binomial random variables such that $X\sim B(2,p)$ and $Y\sim B(4,p)$. If $P(X\ge 1)=5/9$, find (i) $P(Y=4)$ and (ii) $V(2X-Y)$.

A continuous random variable $X$ has uncountably many values in the interval $[a,b]$. If $C$ is a value in the interval $[a,b]$, then $P(X=C)$ is:

Let $A$ and $B$ be two events such that $P(\overline{A\cup B})=\frac{1}{6}$, $P(A\cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{1}{4}$. Then, events $A$ and $B$ are:

A pair of fair dice is thrown. What is the probability of getting a six on both, given at least one die gets six?

The mean of the numbers obtained on throwing a die having written $5$ on four faces, $4$ on three faces, $3$ on two faces and $2$ on one face is?