Probability practice · Medium

An unbiased die is tossed until a number greater than 4 appears. The probability that an even number of tosses is needed, is

Two persons $A$ and $B$ take turns in throwing a pair of dice. The first person to through 9 from both dice will be awarded the prize. If $A$ throws first, then the probability that $B$ wins the game, is

Consider the polynomial $P(x)=x^3+a x^2+b x+c$ where $a, b, c \in\{1,2, \ldots \ldots . ., 10\}$ then the probability that $P(x)$ is divisible by $x^2+x+1$ is

A sample space consists of 3 sample points with associated probabilities given as $2 p, p^2, 4 p-1$. Then the value of $p$ is -

The probability of event $A$ occurring is $0.5$ and B occurring is 0.3. If $A$ and $B$ are mutually exclusive events, then the probability of neither $A$ nor $B$ occurring is

A fair coin is tossed repeatedly until a head is obtained. If $\mathrm{X}$ denotes the number of tosses required, then the conditional probability of $X \geq 6$, given that $X>3$ is

A group consists $2 n$ students including you and your friend. If the group is split into two different sections $A$ and $B$, each containing $n$ students, then the probability that you and your friend are in the different sections is

10 different books and 2 different pens are given to 3 boys so that each gets equal number of things. The probability that the some boy does not receive both the pens is

A five digit mumber is formed by the digits 1, 2, 3, 4, 5,6 and 8. The probability that the number has even digit at both ends is

There are 7 seats in a row. Three persons take seats at random. The probability that the middle seat is always occupied and no two persons are consecutive is

The number of ways in which a pack of 52 cards of four different suits can be distributed equally among four players so that each player gets the Ace, King, Queen and Knave of the same suit is

Two cards are drawn from a well shuffled deck of 52 cards. The probability that one is red card and the other is a queen is

If the papers of 4 students can be checked by any one of the 7 teachers, then the probability that all the 4 papers are checked by exactly 2 teachers is

In the above question the probability that the binary operation is non-commutative is

If $E$ and $F$ are events such that ${P}({E})=\frac{1}{4}, {P}({F})=\frac{1}{2}$ and $P(E$ and $F)=\frac{1}{8}$, then,
I. ${P}({E}\text{ or }{F})=\frac{5}{8}$
II. ${P}(\text{not }{E} \text{ and not } {F})=\frac{3}{8}$

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

Suppose $f\left( x \right) = {x^3} + a{x^2} + bx + c,$ where $a,\,b,\,c$ are chosen respectively by throwing a dice three times. Then, the probability that $f\left( x \right)$ is an increasing function, is

The probability that in the random arrangement of the letters of the word 'UNIVERSITY', the two I's does not come together is

$A$ and $B$ are two students. Their chances of sloving a problem correctly are $\frac{1}{3}$ and $\frac{1}{4}$, respectively. If the probability of their making a common error is, $\frac{1}{20}$ and they obtain the same answer, then the probability of their answer to be correct is

Consider the system of equations $\mathrm{px}+\mathrm{qy}=0, \mathrm{rx}+\mathrm{sy}=0$ where $p, q, r, s \in\{-1,1\}$
Statement-1: The probability that the system of equation has a unique solution is $\frac{1}{2}$.
Statement-2: The probability that the system of equations no solution is zero.

Suppose that the probability that an item produced by a particular machine is defective equals 0.2. If 10 items produced from this machine are selected at random, the probability that not more than one defective is found is

A bag contains 6 red and 3 white balls. Four balls are drawn one by one and not replaced. The probability that they are alternatively of different colours, is

The letters of the word 'longest' are arranged at random. The probability that the vowels may occupy only odd positions, is

Three rifle-men take one shot each at the same target. The probability of the first rifle-man hitting the target is 0.4 , the probability of the second rifle-man hitting the target is 0.5 and the probability of the third rifle-man hitting the target is 0.8 . The Probability that exactly two of them hit the target, is

A man takes a step forward with probability $0.4$ and backwards with probability 0.6. The probability that at the end of eleven steps he is just one step away from the starting point, is

Fifteen coupons are numbered 1 to 15. Seven coupons are selected at random, one at a time with replacement. The probability that the largest number appearing on a selected coupon be 9, is

A group of 6 boys and 6 girls is randomly divided into two equal groups. The probability that each group contains 3 boys and 3 girls 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 $P$ and $Q$ are disjoint sets, is

The probabilities that a student will obtain grades $A,\,B,\,C\;\,\mathrm{or}\;\,D$ are 0.30, 0.35, 0.20 and 0.15 respectively. The probability that he will receive atleast $C$ grade, is

India plays two matches each with the West Indies and Australia. In any match, the probabilities of India getting 0,1 and 2 points are 0.45,0.05 and 0.50, respectively. Assuming that the outcomes are independent, the probability of India getting at least 7 points, is