Probability practice · Mixed difficulty

For a binomial variate $X$ with $n = 6$, if $P\left( {X = 2} \right) = 9P\left( {X = 4} \right)$, then its variance is

If $X$ has a binomial distribution, $B(n,p)$with parameters $n$ and $p$ such that $P\left( {X = 2} \right) = P\left( {X = 3} \right),$ then $E\left( X \right),$ the mean of variable $X,$ 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

If $E$ and $F$ are two events associated with a random experiment for which $P(E)=0.6, P(E \cup F)$ $=0.85$ and $P(E \cap F)=0.42$, then $P(F)$ is

To open a lock, a key is taken out from a collection of $n$ keys at random. If the lock is not opened with this key, it is put back into the collection and another key is tried. The process is repeated again and again. If it is given that with only one key in the collection, the lock can be opened, then the probability that the lock will open in $n$ trials, is

The probability that in the toss of two dice, we obtain the sum 7 or 11, is

A four digit number (numbered from 0000 to 9999) is said to be lucky if the sum of its first two digits is equal to the sum of its last two digits. If a four digit number is picked up at random, then the probability that it is lucky is

The probability that randomly selected calculator from a store is of brand $r$ is proportional to $r$, $r=1,2, \ldots, 6$. Further, the probability of a calculator of brand $r$ being defective is $\frac{7-r}{21}, r=1,2, \ldots, 6$. Then the probability that a calculator randomly selected from the store being defective 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

There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is

A random variable $X$ has the following probability distribution
$\begin{array}{l} X\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,4\\ P(X)\,\,\,k\,\,\,\,\,\,2k\,\,\,\,\,3k\,\,\,\,\,4k \end{array}$
Then, the mean of $X$ is

From a box containing 20 tickets marked with numbers 1 to 20, four tickets are drawn one by one. After each draw, the ticket is replaced. The probability that the largest value of tickets drawn is 15 is

If $P(A \cap B)=\frac{7}{10}$ and $P(B)=\frac{17}{20}$, then $P(A / B)$ equals to

In a maths paper there are 3 sections A, B and C. Section A is compulsory. Out of sections $B$ and $C$ a student has to attempt any one. Passing in the paper means passing in $\mathrm{A}$ and passing in $\mathrm{B}$ or $\mathrm{C}$. The probability of the student passing in $\mathrm{A}, \mathrm{B}$ and $C$ are $p, q$ and $1 / 2$ respectively. If the probability that the student is successful is $1 / 2$ then

Probability that in the toss of two dice we obtain an even sum or a sum less than 5, is

Assuming that for a husband-wife couple the chances of their child being a boy or a girl are the same, the probability of their two children being a boy and a girl is

For $k = 1,2,3$ the box ${B_k}$ contains $k$ red balls and $\left( {k + 1} \right)$ white balls,
Let $P\left( {{B_1}} \right) = \frac{1}{2},\,P\left( {{B_2}} \right) = \frac{1}{3}$ and $P\left( {{B_3}} \right) = \frac{1}{6}$
A box is selected at random and a ball is drawn from it. If a red ball is drawn, then the probability that it has come from box ${B_2}$, 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

A natural number $x$ is chosen at random from the first 100 natural numbers. The probability that $x + \frac{{100}}{x} > 50$

If $X$ is a poisson variate such that $P\left( {X = 1} \right) = P\left( {X = 2} \right)$, then $P\left( {X = 4} \right)$ is equal to

Four tickets are marked 00,01, 10 and 11 respectively, are placed in a bag. A ticket is drawn at random five times, being replaced each time. The probability that the sum of the numbers on the five tickets drawn is 24, is

There are 12 white and 12 red balls in a bag. Balls are drawn one by one with replacement from the bag. The probability that 7 th drawn ball is 4 th white, is

If $P(B)=\frac{3}{5}, P(A / B)=\frac{1}{2}$ and $P(A \cup B)=\frac{4}{5}$, then $P(A \cup B)^{\prime}+P\left(A^{\prime} \cup B\right)$ is equal to

Two numbers $a$ and $b$ are chosen at random from the set $\{1,2,3, \ldots, 5 n\}$. The probability that $a^4-b^4$ is divisible by 5, is

In tossing 10 coins, the probability of getting exactly 5 heads is

Suppose $A$ and $B$ are two events such that $P(A)>0$ and $P(B) \neq 1$, then $P\left(A^{\prime} \mid B^{\prime}\right)$ equals

Let $x$ be a non-zero real number. A determinant is chosen from the set of all determinants of order 2 with entries $x$ or $-x$ only. The probability that the value of the determinant is non-zero is

A bag contains $a$ white and $b$ black balls. Two players $A$ and $B$ alternately draw a ball from the bag replacing the ball each time after the draw till one of them draws a white ball and wins the game. $A$ begins the game. If the probability of $A$ winning the game is three times that of $B$, then the ratio $a:b$ 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

Which one is not a requirement of a binomial distribution?