Embibe Experts Solutions for Chapter: Probability, Exercise 3: Level 3

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

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______.

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 ________.

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

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 :

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

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 ${\log }_{10} 2=0.3010, {\log }_{10} 3=0.4771$)