Probability practice · Medium

Consider 5 independent Bernoulli's trials each with probability of success $P$. If the probability of atleast one failure is greater than or equal to $\frac{31}{32}$, then $P$ lies in the interval

The chance of an event happening is the square of the chance of a second event but the odds against the first are the cube of the odds against the second. The chances of the events are

The probability that in a random arrangement of the letters of the word ‘UNIVERSITY’, the two $I'\,\mathrm{s}$ do not come together is

Two numbers $x$ and $y$ are chosen at random from the set $\{1,2,3, \ldots, 30\}$. The probability that $x^2-y^2$ is divisible by 3 is

Let $0 < P\left( A \right) < 1,0 < P\left( B \right) < 1$ and $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( A \right)\;P\left( B \right).$ Then,

The probability that at least one of the events $A$ and $B$ occurs is 0.7 and they occur simultaneously with probability 0.2. Then, $P\left( {\bar A} \right) + P\left( {\bar B} \right) =$

If the probability density function of a random variable $X$ is $f\left( x \right) = \frac{x}{2}$ in $0 \le x \le 2,$ then $P\left( {\frac{{X > 15}}{{X > 1}}} \right)$ is equal to

If $a$ is an integer lying in $[-5,30]$, then the probability that the graph of $y=x^2+2(a+4) x$ $-5 a+64$ is strictly above the $x$-axis is

Probability of throwing 16 in one throw with three dice is

${A}, {B}$ and ${C}$ are three mutually exclusive and exhaustive events. If $\frac{1}{3} {P}({C})=\frac{1}{2} {P}({A})={P}({B})$, then ${P}({B})=$

10 persons sit around a circular table with 10 numbered chairs. The probability that the two particular persons $A$ and $B$ are always together is

The odds in favour of a book reviewed by three independent critics are, respectively, $5: 2,4: 3$ and $3: 4$. The probability that majority of the critics give favourable remark is

$\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \ldots \ldots \mathrm{x}_{50}$ are fifty real numbers such that $x_r < x_{r+1}$ for $r=1,2,3, \ldots .49$. Five numbers out of these are picked up at random and they are arranged in increasing order. The probability that the five numbers have $x_{20}$ as the middle number is :

A contest consists of predicting the results (win, draw or defeat) of 10 football matches. The probability that one entry contains at least 5 correct answers is

Two persons each make a single throw with a pair of dice. The probability that the throws are equal, is

If three students $A,\,B,\,C$ can solve a problem with probabilities $\frac{1}{3},\,\frac{1}{4}$ and $\frac{1}{5}$ respectively, then the probability that the problem will be solved is

Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^N < N !$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $4 m-3 n$ is equal to

A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing on the fifth toss, equals

Let a die be rolled $n$ times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $\frac{k}{2^{15}}$, then $k$ is equal to

Four persons are chosen at random from a group of 3 men, 2 women and 4 children. The chance that exactly 2 of them are children, is

In a college 25% boys and 10% girls offer Mathematics. There are 60% girls in the college. If a Mathematics student is chosen at random, then the probability that the student is a girl, will be

A bag contains a mixed lot of red and blue balls. If two balls are drawn at random, the probability of drawing two red balls is five times the probability of drawing two blue balls and the probability of drawing one ball of each colour is six times the probability of drawing two blue balls. Then, the total number of red and blue balls in the bag is

Sum of two positive integers is 66 and $\mu$ is the maximum value of their product $S=\left\{x \in \mathbb{Z}, x(66-x) \geq \frac{5 \mu}{9}\right\}, x \neq 0$, then probability of $A$ when $A=\{x \in S ; x=3 k, x \in \mathbb{N}\}$ is:

From a group of 8 boys and 3 girls, a committee of 5 members to be formed. Find the probability that 2 particular girls are included in the committee

In a sequence of independent trials, the probability of success is $1 / 4$. If $p$ denotes the probability that the second success occurs on the fourth trial or later trial, then $32 p-20$ is

A complete cycle of a traffic light takes 60 s. During each cycle the light is green for 25 s, yellow for 5 s and red for 30 s. At a randomly chosen time, the probability that the light will not be green, is

An experiment succeeds twice as often as it fails. Find the probability that in 4 trials there will be at least three success

There are 10 pairs of shoes in a cupboard, from which 4 shoes are picked at random. If $p$ is the probability that there is at least one pair, then $323 p$ -97 is equal to

If $A$ and $B$ are events such that $P(A)=0.42, P(B)=0.48$ and $P(A \text{ and } B)=0.16$, then,

I. ${P}(\operatorname{not}{A})=0.58$

II. ${P}(\text{not }{B})=0.52$

III. ${P}({A} \text{ or }{B})=0.47$

Seven chits are numbered 1 to 7. Four chits are drawn one by one with replacement. The probability that the least number appearing on any selected chit is 5, is