Probability

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 _____.

Two players, $P_1$ and $P_2$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P_1$ and $P_2$, respectively. If $x > y$, then $P_1$ scores 5 points and $P_2$ scores 0 point. If $x = y$, then each player scores 2 points. If $x < y$, then $P_1$ scores 0 point and $P_2$ scores 5 points. Let $X_i$ and $Y_i$ be the total scores of $P_1$ and $P_2$, respectively, after playing the $i^{th}$ round.

The correct option is:

Suppose that

• Box-I contains 8 red, 3 blue and 5 green balls,
• Box-II contains 24 red, 9 blue and 15 green balls,
• Box-III contains 1 blue, 12 green and 3 yellow balls,
• Box-IV contains 10 green, 16 orange and 6 white balls.

A ball is chosen randomly from Box-I; call this ball $b$. If $b$ is red then a ball is chosen randomly from Box-II, if $b$ is blue then a ball is chosen randomly from Box-III, and if $b$ is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to

Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and $n$ white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is $\dfrac6{11}$, then $n$ is equal to ______.

A random variable $X$ has the following probability distribution:
$$\begin{array}{|c|c|c|c|c|c|}\hline X&0&1&2&3&4\\\hline P(X)&k&2k&4k&6k&8k\\\hline\end{array}$$
The value of $P(1<X<4\mid X\le2)$ is equal to :

Let $E_1$ and $E_2$ be two events such that the conditional probabilities $P(E_1|E_2)=\dfrac12$, $P(E_2|E_1)=\dfrac34$ and $P(E_1\cap E_2)=\dfrac18$. Then :

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

The probability distribution of X is :

X	0	1	2	3
P(X)	$\frac{1-d}4$	$\frac{1+2d}4$	$\frac{1-4d}4$	$\frac{1+3d}4$

For the minimum possible value of d, sixty times the mean of X 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 :

If A and B are two events such that $P(A)=\dfrac13$, $P(B)=\dfrac15$ and $P(A\cup B)=\dfrac12$, then $P(A|B')+P(B|A')$ is equal to

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 :