Probability practice · Mixed difficulty

Let $A$ be a set consisting of $n$ elements. The probability of selecting two subsets $P$ and $Q$ of set $A$ such that $Q=\bar{P}$, is

Let a random variable X have a binomial distribution with mean 8 and variance 4.
If $P\left( {X \le 2} \right) = \frac{k}{{{2^{16}}}},$ then $k$ is equal to

The numbers $1,2,3, \ldots$, $n$ are arranged in a random order. The probability that the digits $1,2,3, \ldots, k(k < n)$ appears as neighbours in that order is

There are 100 students in a collage class of which 36 are boys studying statistics and 13 girls not studying statistics. If there are 55 girls in all, the Probability that a boy picked up at random is not studying, is

Difference between mean and variance of a binomial variate is '1' and difference between their square is '11'. Then probability of getting exactly three success is:

An experiment has 10 equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is

A person draws a card from a pack, replaces it, shuffles the pack, again draws a card, replaces it and draws again. This he does until he draws a heart. The probability that he will have to make at least four draws is

If $A$ and $B$ are two events, then $P$ (neither $A$ nor $B$) equals

If $P\left( {A \cup B} \right) = 3/4$ and $P\left( {\bar A} \right) = 2/3,$ then $P\left( {\bar A \cap B} \right)$ is equal to

If out of $2 \mathrm{n}$ tickets, numbered as $1,2,3, \ldots 2 \mathrm{n}, 3$ distinct are chosen at randomly, and probability that they are in A.P. is $\frac{1}{10}$, then &#39; $n$ &#39; is

A biased coin with probability $p, 0 < p < 1$, of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $2 / 5$, then $p$ is equal to

If $A$ and $B$ are two events than the value of the determinant choosen at random from all the determinants of order 2 with entries 0 or 1 only is positive or negative respectively. Then,

Six boys and six girls sit in a row randomly. The probability that boys and girls sit alternatively is

Let $A$ be a set of all 4-digit natural numbers whose exactly one digit is 7. Then, the probability that a randomly chosen element of $A$ leaves remainder 2 when divided by 5 is

A carton contains 20 bulbs, 5 of which are defective. The probability that, if a sample of 3 bulbs is chosen at random from the carton, 2 will be defective, is

A six faced die is biased such that $3 \times P$ (a prime number) $=6 \times P$ (a composite number) $=2 \times P(1).$

Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $X$ is :

Five persons entered the lift cabin on the ground floor of an 8-floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first. The probability of all five persons leaving at different floors, is

An experiment yields 3 mutually exclusive and exhaustive events $A,B,C.$ If $P\left( A \right) = 2P\left( B \right) = 3P\left( C \right),$ then $P\left( A \right)$ is equal to

Let $A, B$ and $C$ be three events such that $P(A)=0.3, \quad P(B)=0.4, \quad P(C)=0.8, \quad P(A \cap B)=0.08$, $P(A \cap C)=0.28, P(A \cap B \cap C)=0.09$. If $P(A \cup B \cup C) \geq 0.75$, then show that $P(B \cap C)$ satisfies

There are $n$ urns each containing $(n+1)$ balls such that the $i^{\text {th }}$ urn contains ' $i$ ' white balls and $(n+1-i)$ red balls. Let $U_i$ be the event of selecting $i^{t h} u r n, i=1,2,3, \ldots, n$ and $W$ denotes the event of getting a white ball. If $P\left(U_i\right) \propto i$, where $i=1,2,3, \ldots, n$, then $\lim\limits_{n \to \infty } P(W)$ is equal to

A biased die is tossed and the respective probabilities for various faces to turn up are
$\begin{array}{llllllllllllllllllll} {Face\;\;\;\;\;\;\;\;\;\;\;:\;1\;\;\;\;\;\;2\;\;\;\;\;\;\;\;3\;\;\;\;\;\;\;4\;\;\;\;\;\;\;5\;\;\;\;\;\;\;6}\\ {Probability\mathrm{ }:\;0.1\;\;\,\,0.24\;\;\,\,0.19\;\,\,0.18\;\,\,0.15\;\,\,\,0.14} \end{array}$
If an even face has turned up, then the probability that it is face 2 or face 4, is

Two numbers are selected randomly from the set $S = \left\{ {1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6} \right\}$ } without replacement one by one. The probability that minimum of the two number is less than 4, is

Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane, is

If the probability of $A$ to fail in an examination is 0.2 and that for $B$ is 0.3, then probability that either $A$ or $B$ is fail, is

If two numbers $p$ and $q$ are choosen randomly from the set $\{1,2,3,4\}$ with replacement, then the probability that $p^2 \geq 4 q$ is equal to

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:

A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability. The smallest value of $n$ , so that the probability of guessing at least $'n'$ correct answer is less than $\frac{1}{2}$ , is

There are $n$ letters and $\mathrm{n}$ addressed envelopes, the probability that all the letters are not kept in the right envelope, is

A letter is taken out at random from ‘ASSISTANT’ and another is taken out from ‘STATISTICS’. The probability that they are same letters, is

If $\mathrm{A}$ and $\mathrm{B}$ are two events such that $\mathrm{A} \subset \mathrm{B}$ and $P(B) \neq 0$, then which of the following is correct: