Classical probability

Let $a,b,c\in\{1,2,3,4\}$. If the probability, that $ax^2+2\sqrt2\,bx+c>0$ for all $x\in\mathbb R$, is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to ________.

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to :

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $p$, then $96p$ is equal to ________.

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\dfrac ab$, where $a,b\in\mathbb N$ and $\gcd(a,b)=1$, then $a+b$ is equal to ________.

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :

A bag contains $N$ balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For $i = 1, 2, 3$, let $W_i$, $G_i$, and $B_i$ denote the events that the ball drawn in the $i^{\text{th}}$ draw is a white ball, green ball, and blue ball, respectively. If the probability $P(W_1 \cap G_2 \cap B_3) = \dfrac{2}{5N}$ and the conditional probability $P(B_3 \mid W_1 \cap G_2) = \dfrac29$, then $N$ equals _____.

Let $X = \left\{(x, y) \in \mathbb{Z}\times\mathbb{Z} : \dfrac{x^2}{8} + \dfrac{y^2}{20} < 1 \text{ and } y^2 < 5x\right\}$. Three distinct points $P$, $Q$ and $R$ are randomly chosen from $X$. Then the probability that $P$, $Q$ and $R$ form a triangle whose area is a positive integer, is

Two distinct points are chosen randomly out of the points $A_1, A_2, \ldots, A_{49}$. Let $p$ be the probability that they are friends. Then the value of $7p$ is _____.

In a study about a pandemic, data of 900 persons was collected. It was found that

• 190 persons had symptom of fever,
• 220 persons had symptom of cough,
• 220 persons had symptom of breathing problem,
• 330 persons had symptom of fever or cough or both,
• 350 persons had symptom of cough or breathing problem or both,
• 340 persons had symptom of fever or breathing problem or both,
• 30 persons had all three symptoms (fever, cough and breathing problem).

If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is _____.

A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with mark $n$ is $\dfrac1n$. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, is :

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is $p$, then $96p$ is equal to ______.

Five numbers $x_1,x_2,x_3,x_4,x_5$ are randomly selected from the numbers $1,2,3,\dots,18$ and are arranged in the increasing order $(x_1<x_2<x_3<x_4<x_5)$. The probability that $x_2=7$ and $x_4=11$ is :

If a point $A(x,y)$ lies in the region bounded by the $y$-axis, straight lines $2y+x=6$ and $5x-6y=30$, then the probability that $y<1$ is

Let $S=\{E_1,E_2,\dots,E_8\}$ be a sample space of a random experiment such that $P(E_n)=\dfrac n{36}$ for every $n=1,2,\dots,8$. Then the number of elements in the set $\left\{A\subseteq S:P(A)\ge\dfrac45\right\}$ is ______.

The probability, that in a randomly selected 3-digit number at least two digits are odd, is

The probability that a randomly chosen one-one function from the set $\{a,b,c,d\}$ to the set $\{1,2,3,4,5\}$ satisfies $f(a)+2f(b)-f(c)=f(d)$ is :

The probability that a randomly chosen $2\times2$ matrix with all the entries from the set of first 10 primes, is singular, is equal to :

The probability that a relation R from $\{x,y\}$ to $\{x,y\}$ is both symmetric and transitive, is equal to

If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that $x^2+\alpha x+\beta>0$, for all $x\in\mathbf R$, is :

Let $E_1$, $E_2$, $E_3$ be three mutually exclusive events such that $P(E_1)=\dfrac{2+3p}6$, $P(E_2)=\dfrac{2-p}8$ and $P(E_3)=\dfrac{1-p}2$. If the maximum and minimum values of p are $p_1$ and $p_2$, then $(p_1+p_2)$ is equal to :