Probability practice · Mixed difficulty

A positive integer $N$ is selected so as to be $100 < N < 200$. Then, the probability that it is divisible by 4 or 7 , is

If $P(A)=1 / 4, P(B)=1 / 2, P(A \cup B)=5 / 8$ then $P(A \cap B)$ is

A binary operation is chosen at random from the set of all binary operations on a set $A$ containing $n$ elements. The probability that the binary operation is commutative, 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): $P\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}$

Given a throw of three dice shows different face, the probability that at least one face shows 6 is

The mean and variance of a binomial distribution are 4 and 3 respectively. Then the probability of getting exactly six successes in this distribution, is

A seven digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed is divisible by 2, is

If $P(E)=0.40, P(F)=0.35, P(E \cup F)=0.55$, then $P(F / E)$ is

If $P(B)=\frac{3}{4}, P(A \cap B \cap \bar{C})=\frac{1}{3}$ and $P(\bar{A} \cap B \cap \bar{C})=\frac{1}{3}$, then $P(B \cap C)$ is

A coin is tossed $n$ times. The probability that head will turn up an odd number of times, is

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: $P(X)+P(Y)=2 P(X \cap Y)$

$\mathrm{A}$ and $\mathrm{B}$ are two independent events. The probability that both $\mathrm{A}$ and $\mathrm{B}$ occur is $1 / 6$ and the probability that neither of them occurs is $1 / 3$. The probability of occurrence of A is:

If $M$ and $N$ are any two events. The probability, that exactly one of them occurs, is

Four digit numbers are formed using each of the digits $1,2, \ldots, 8$ only once. One number from them is picked up at random. The probability that the selected number contains unity is

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

A bag contains $w$ white and $b$ black balls. A ball is drawn at random and put back in the bag with $k$ additional balls of the same colour. A ball is again drawn at random from the bag. The probability that it is black 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

If $C$ and $D$ are two events, such that $C \subset D$ and $P(D) \neq 0$, then the correct statement among the following, is

A single letter is selected at random from the word ‘PROBABILITY’. The probability that it is a vowel is

Suppose $A, B, C$ are three independent events such that $P(A)=P(B)=P(C)=p$, then $P$ (at least two of $A, B, C$) equals

The probability that an electric bulb will last 150 days
or more is 0.7 and that it will last at most 160 days
is 0.8. The probability that the bulb will last 150 to
160 days is

One ticket is selected at random from 50 tickets numbered 00,01,02,...,49. Then the probability that the sum of the digits on the selected ticket is 8 , given that the product of these digits is zero, equals

Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by one without replacement. The probability that at least one ball is red, is

A cricket club has 15 members, of whom only 5 can bowl. If the names of 15 members are put into a box and 11 are drawn at random, then the probability of obtaining an eleven containing at least 3 bowlers is

If two natural number $\mathrm{x}, \mathrm{y}$ are selected at randomly and probability that $x^2+y^2$ is multiply of 5 is $p$, then $25 \mathrm{p}$ 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

A, B, C try to hit a target simultaneously but independently. Their respective probabilities of hitting the targets are $\frac{3}{4}, \frac{1}{2}, \frac{5}{8}$. The probability that the target is hit by $\mathrm{A}$ or B but not by $\mathrm{C}$ is :

Three dice are thrown. The probability of getting a sum which is a perfect square, 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