Probability practice · Mixed difficulty

In a party 23 persons take their seats at a round table. The odds against two particular persons sitting together are:

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:

A man is known to speak truth in 75% cases. If he throws an unbiased die and tells his friend that it is a six, then the probability that it is actually a six, is

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

An article manufactured by a company consists of two parts ${X}$ and ${Y}$. In the process of manufacture of part ${X}$, 9 out of 104 parts may be defective. Similarly, 5 out of 100 are likely to be defective in the manufacture of the part $Y$. The probability that the assembled product will not be defective, is

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

A pack of cards contains 4 aces, 4 kings, 4 queens and 4 jacks. Two cards are drawn at random. The probability that atleast one of them is an ace is

The probability that the roots of the equation ${x^2} + nx + \frac{1}{2} + \frac{n}{2} = 0$ are real where $n \in N$ such that $n \le 5,$ is

Box-I contains 30 cards numbered 1 to 30 and Box - II contains cards numbered 31 to 50 . A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box - I is

In a binomial distribution $B(n, p)$, the sum and the product of the mean and the variance are 5 and 6 respectively, then $6(n+p-q)$ is equal to

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

If $X$ follows a binomial distribution with parameters $n = 6$ and $p.$ If $4\left( {P\left( {X = 4} \right)} \right) = P\left( {X = 2} \right),$ then $p =$

The digits $1,2,3,4,5,6,7,8$ and 9 are written in random order to form a nine digit number. The probability that this number is divisible by 4, is

Out of 40 consecutive natural numbers, two are chosen at random. Probability that the sum of the numbers is odd, is

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is $\frac{4}{5}$, then the probability that he is unable to solve less than two problem is

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

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of 6 before $B$ throws a total of 7 and $B$ wins the game if he throws a total of 7 before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is

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

A bag contains 4 red and 4 blue balls. Four balls are drawn one by one from the bag, then find the probability that the drawn balls are in alternate colour.

In a book of 500 pages, it is found that there are 250 typing errors. Assume that poisson law holds for the number of errors per page. Then, the probability that a random sample of 2 pages will contain no error, is

Three persons $A, B$ and $C$ are to speak at a function along with 7 other persons. The probability that $A, B$, and $C$ speak together

A box contain $N$ coins, $m$ of which are fair and rest are biased. The probability of getting a head when a fair coin is tossed is $1 / 2$, while it is $2 / 3$ when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows head and the second time it shows tail. The probability that the coin drawn is fair is

Three distinct numbers are selected from first 100 natural numbers. The probability that all the three numbers are divisible by 2 and 3 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

Suppose two numbers $a$ and $b$ are chosen randomly from the set $\{1,2,3,4,5,6\}$. The probability that function $f(x)=x^3+a x^2+b x$ is strictly increasing function on $\mathbf{R}$ is:

$A$ and $B$ play a game where each is asked to select a number from 1 to 25. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial, is

One coin is thrown 100 times. The probability of coming tail in odd number

A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to probability of getting 9 heads, then the probability of getting 2 heads is.

The mean and variance of a binomial distribution are 6 and 4. The parameter $n$ is

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