Probability practice · Medium

A number $x$ is chosen at random from the set $\{1,2,3,4, \ldots$, $100\}$. Define the event: $\mathrm{A}=$ the chosen number $\mathrm{x}$ satisfies $\frac{(\mathrm{x}-10)(\mathrm{x}-50)}{(\mathrm{x}-30)} \geq 0$
Then P (A) is:

There are $n$ stations between two cities $A$ and $B$. A train is to stop at three of these $n$ stations. What is the probability that no two of these three stations are consecutive?

A class contains 20 boys and 20 girls of which half the boys and half the girls have cat eyes. If one student is selected from the class & if the probability that either the student is a boy or has cat eyes is $\frac{{a}}{{b}}$, then $\sqrt{{a}^2+{b}^2}=$

$A$ and $B$ are are two events. Odds against $A$ are 2 to 1 and odds in favour of $A \cup B$ are 3 to1. If $x \le P\left( B \right) \le y$, then ordered pair $\left( {x,y} \right)$ is

One hundred identical coins, each with probability, p, of showing up heads are tossed once. If $0 < p < 1,$ and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of $p,$ is

A car is parked by an owner amongst 25 cars in a row, not at either end. On his return he finds that exactly 15 places are still occupied. The probability that both the neighboring places are empty is

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is

An almirah stores 5 black and 4 white socks well mixed. A boy pull out 2 socks at random. The probability that 2 are of the same colour is

If the letters of the word ‘REGULATION’ be arranged at random, the probability that there will be exactly 4 letters between $R$ and $E$ is

$A$ is a set containing $n$ elements. A subset $P$ of $A$ is chosen at random. The set $A$ is reconstructed by replacing the elements of $P$. A subset $Q$ is again chosen at random. The probability that $P \cup Q$ contains just one element, is

Suppose $A$ and $B$ are two mutually exclusive events such that $P(A)=\frac{1}{3}(3 p+1)$ and $P(B)=\frac{1}{4}(1-p)$, then the set of possible values of $p$ lies in the interval:

An unbiased cubical die is thrown $n$ times. If $x_i=$ outcome on the $i^{\text {th }}$ throw and $X=\operatorname{gcd}\left(x_1, x_2, \ldots . ., x_n\right)$, then $E(X)$ is equal to

Nine balls of the same size and color, numbered $1,2, \ldots . ., 9$, were put into a packet. Now A draws a ball from the packet, noted that it is of number a, and puts back it. Then B also draws a ball from the packet and noted that it is of number $b$. Then the probability for the inequality $a-2 b+$ $10>0$ to hold is

The probability that in a year of the 22nd century chosen at random there will be 53 Sundays, is

A random variate $X$ takes the values 0, 1, 2, 3 and its mean is 1.3. If $P\left( {X = 3} \right) = 2P\left( {X = 1} \right)$ and $P\left( {X = 2} \right) = 0.3$, then $P\left( {X = 0} \right)$ is equal to

If the letters of the word ‘REGULATIONS’ be arranged at random, the probability that there will be exactly 4 letters between $R$ and $E$ is

There are 4 white and 3 black balls in a box. In another box there are 3 white and 4 black balls. An unbiased dice is rolled. If it shows a number less than or equal to 3 then a ball is drawn from the first box but if it shows a number more than 3 then a ball is drawn from the second box. If the ball drawn is black then the probability that the ball was drawn from the first box is

If $P\left( {A \cup B} \right) = 0.8$ and $P\left( {A \cap B} \right) = 0.3,$ then $P\left( {A'} \right) + P\left( {B'} \right)$ equals to

A person throws two dice, one the common cube and the other regular tetrahedron with numbers $1,2,3,4$ on its faces, the number on the lowest face being taken in the case of a tetrahedron. The chance that the sum of numbers thrown is not less than 5 is

If $A$ and $B$ are events such that $P(\bar{A} \cup \bar{B})=\frac{3}{4}, \quad P(\bar{A} \cap \bar{B})=\frac{1}{4}$ and $P(\bar{A})=\frac{2}{3}$, then $P(\bar{A} \cap B)$ is

${A}, {B}$ are two events in a random experiment such that $0 < p(A) < 1$ and $P(B) \neq 1$
Assertion (A): $P\left(\frac{A}{B^c}\right)=\frac{P(A)-P(A \cap B)}{1-P(B)}$
Reason (R): $P\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}$

Let $X$ and $Y$ are two events such that $P(X \cup Y)=P(X \cap Y)$.
Statement 1: $P\left(X \cap Y^{\prime}\right)=P\left(X^{\prime} \cap Y\right)=0$
Statement 2: $P(X)+P(Y)=2 P(X \cap Y)$

A coin is biased so that a head is twice as likely to
occur as a tail. If the coin is tossed 3 times, the
probability of getting two tails and one head is

Three six-faced dice are thrown together. The probability that the sum of the numbers appearing on the dice is $k(3 \leq k \leq 8)$, is

A determinant of the second order is made with the elements 0 and 1. If $\frac{{m}}{{n}}$ be the probability that the determinant made is non-negative, where ${m}$ and ${n}$ are relative primes, then the value of ${n}-{m}$ is

Out of 400 people, 160 are smokers and non-vegetarian, 100 are vegetarian smokers, 140 are non-smokers and vegetarian. If a person is smoker and non-vegetarian, then the chances of his death is $35 \%$, if he is vegetarian and smoker then chances of his death is $15 \%$. If he is non-smoker and vegetarian then chances of his death is $10 \%$. If a persons is found to be dead, then the probability that he is smoker and non-vegetarian, is

If the integers $m$ and $n$ are chosen at random between 1 and 100, then the probability that a number of the form ${7^m} + {7^n}$ is divisible by 5, equals

Two dice are thrown independently. Let $A$ be the event that the number appeared on the $1^{\text {st }}$ die is less than the number appeared on the $2^{\text {nd }}$ die, $B$ be the event that the number appeared on the $1^{\text {st }}$ die is even and that one the second die is odd, and $C$ be the event that the number appeared on the $1^{\text {st }}$ die is odd and that on the $2^{\text {nd }}$ is even. Then

Three randomly chosen non negative integers $x, y$ and $z$ are found to satisfy eq. $x+y+z=10$ and the probability that $z$ is even is $p / q$ ($p, q$ coprime) then ${p}+{q}=$ ____

The probability that a certain kind of component will survive a given shock test is $\frac{3}{4}$. The probability that exactly 2 of the next 4 components tested survive is