Embibe Experts Solutions for Chapter: Probability, Exercise 2: Exercise

Let $p, q$ be chosen one by one from the set $\left\{1,\sqrt{2},\sqrt{3},2,e,\pi \right\}$ with replacement. Now a circle is drawn taking $\left(p,q\right)$ as its centre. Then the probability that at the most two rational points exist on the circle is (rational points are those points whose both the coordinates are rational)

A car is parked among $N$ car standing in a row, but not at either end. On his return, the owner finds that exactly $r$ of the $N$ places are still occupied. The probability that the places neighbouring his car are empty is

One ticket is selected at random from $100$ tickets numbered $00, 01, 02, \dots , 99$. Suppose $A$ and $B$ are the sum and product of the digits found on the ticket. Then $P\left(\frac{\left(A=7\right)}{\left(B=0\right)}\right)$ is given by

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

Consider $f\left(x\right)=x^3+a{x}^2+bx+c$. Parameters $a, b, c$ are chosen respectively by throwing a die three times, then the probability that $f\left(x\right)$ is an increasing function is

An unbiased die, with faces numbered $1, 2, 3, 4, 5, 6$ is thrown $n$ times and the list of $n$ numbers showing up is noted. Then the probability that, among the numbers $1, 2, 3, 4, 5, 6,$ only three numbers appear in this list?

A boy has $20\%$ chance of hitting at a target. Let $p$ denote the probability of hitting the target for the first time at the $n^{\mathrm{th}}$ trial. If $p$ satisfies the inequality $625p^2-175p+12<0,$ then value of $n$ is

STATEMENT-$1$: Since sample space of the experiment 'A coin is tossed if it turns up head, a die is thrown' is $\left\{\left(H,1\right), \left(H,2\right), \left(H,3\right), \left(H,4\right), \left(H,5\right), \left(H,6\right), T\right\}$.

$\therefore$Probability of the event $\left\{\left(H,1\right), \left(H,2\right), \left(H,5\right)\right\}$ is $\frac{3}{7}$.

STATEMENT-$2$: If all the sample points in the sample space of an experiment are pair wise mutually exclusive, equally likely and exhaustive, then probability of an event $E$ is defined as

$P\left(E\right)=\frac{ \text{number of sample points favourable to the event} E}{ \text{Total number of sample points in the sample space} }$