Level 3

Box-I contains $3$ cards bearing numbers $1,2,3$; Box II contains $5$ cards bearing numbers $1,2,3,4,5$ and Box III contains $7$ cards bearing numbers $1,2,3,4,5,6,7$. One card is drawn at random from each of the boxes. If $x_i$ be the number on the card drawn from the ${\mathrm{i}}^{\mathrm{th} }$ box, $i=1,2,3$ then the probability that $x_1+x_2+x_3$ is odd is equal to

Let there be three independent events $E_1,E_2$ and $E_3.$ The probability that only $E_1$ occurs is $\alpha$ only $E_2$ occurs is $\beta$ and only $E_3$ occurs is $\gamma .$ Let ${}^{\text{'}}p^{\text{'}}$ denote the probability of none of events occurs that satisfies the equations $(\alpha -2\beta )\mathrm{p}=\alpha \beta$ and $(\beta -3\gamma )\mathrm{p}=2\beta \gamma .$ All the
given probabilities are assumed to lie in the interval $\left(0, 1\right).$

Then, $\frac{\text{ Probability of occurrence of }{E}_1}{\text{ Probability of occurrence of }{E}_3}$ is equal to ________.

Let $B_i\left(i=1,2,3\right)$ be three independent events in a sample space. The probability that only $B_1$ occur is $\alpha ,$ only $B_2$ occurs is $\beta$ and only $B_3$ occurs is $\gamma .$ Let $p$ be the probability that none of the events $B_i$ occurs and these $4$ probabilities satisfy the equations $\left(\alpha -2\beta \right)p=\alpha \beta$ and $\left(\beta -3\gamma \right)p=2\beta \gamma$ (All the probabilities are assumed to lie in the interval $\left.\left(0,1\right)\right)$ Then $\frac{P\left(B_1\right)}{P\left(B_3\right)}$ is equal to______.

In a group of $400$ people, $160$ are smokers and non-vegetarian; $100$ are smokers and vegetarian and the remaining $140$ are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35\%,20\%$ and $10\%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :

Let $A$ be a set containing $n$ elements. A subset $P$ of the set $A$ is chosen at random. The set $A$ is reconstructed by replacing the elements of $P,$ and another subset $Q$ of $A$ is chosen at random. The probability that $P\cap Q$ contains exactly $m\left(m<n\right)$ elements is

A bag contains $20$ coins. If the probability that the bag contains exactly $4$ biased coin is $\frac{1}{3}$ and that of exactly $5$ biased coin is $\frac{2}{3}$, then the probability that all the biased coin are sorted out from the bag in exactly $10$ draws is

Two aeroplanes I and II bomb a target in succession. The probability 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

Find the minimum number of tosses of a pair of dice, so that the probability of getting the sum of the numbers on the dice equal to $7$ on atleast one toss, is greater than $0.95$. (Given ${\log }_{10} 2=0.3010, {\log }_{10} 3=0.4771$)

The probability mass function of a random variable $X$ is

Then the standard deviation of $X$ is

Five defective mangoes are accidently mixed with $15$ good ones. Four mangoes are drawn at random from this lot. Then, the probability distribution of the number of defective mangoes is

The mean and standard deviation of random variable $X$ are $10$ and $5$ respectively, then $E{\left(\frac{X-15}{5}\right)}^2=$

A random variable $X$ takes the value $1, 2, 3$ and $4$ such that $2P(X=1)=3P\mathit{(}X=2)=P(X=3)=5P(X=4)$. If ${\sigma }^2$ is the variance and $\mu$ is the mean of $X$, then ${\sigma }^2+{\mu }^2=$

In four schools $B_1\mathit{,}\mathit{ }{B}_{2 }\mathit{,}\mathit{ }{B}_3\mathit{,}\mathit{ }{B}_4$ the percentage of girls students is $12, 20, 13, 17$, respectively. From a school selected at random, one student is picked up at random, and it is found that the student is a girl. The probability that the school selected is $B_2$, is

In a hurdle race, a runner has probability $p$ of jumping over a specific hurdle. Given that in $5$ trials, the runner succeeded $3$ times, the conditional probability that the runner had succeeded in the first trial, is

Let a function $\mathrm{f}:X\to Y$ be defined where $X=\{0,1,2,3,\dots 9$$\}$$,Y=\{0,1,2,\dots .,100\}$ and $\mathrm{f}\left(5\right)=5,$ then the probability that the function of the type $\mathrm{f}:X\to B$ where $B\subseteq Y$  is bijective in nature is

Matrices of order $3\times 3$ are formed by using the elements of the set $\mathrm{A}=\{-3,-2,-1,0,1,2,3\},$ then probability that matrix is either Symmetric or Skew Symmetric, is

Two squares of $1\times 1$ are chosen at random on a chessboard. What is the probability that they have a side in common?