G Tewani Solutions for Chapter: Probability I, Exercise 1: Exercises

Let $E$ be an event which is neither a certainty nor an impossibility. If probability is such that $P\left(E\right)=1+\lambda +{\lambda }^2$ and $P\left(E^{\text{'}}\right)={\left(1+\lambda \right)}^2$ in terms of an unknown $\lambda$, then $P\left(E\right)$ is equal to

A natural number $x$ is chosen at random from the first $100$ natural numbers. The probability that $x+\frac{100}{x}>50$ is

If $a\in I$ and $a\in \left[-5,30\right]$, then the probability that the graph of $y=x^2+2\left(a+4\right)x-5a+64$ is strictly above the $x$-axis is

Forty teams play a tournament. Each of them plays with every other team just once. Each game result is a win for one team. If each team has a $50\%$ chance of winnings each game, the probability that at the end of the tournament, every team has won a different number of games is

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

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