Basics of Probability

Before a race the chance of three runners $A, B$ & $C$ were estimated to be proportional to $5, 3 \& 2$ respectively but during the race $A$ meets with an accident which reduces his chance to $\frac{1}{3}$. If the respective chances of $B$ and $C$ are $P\left(B\right)$ and $P\left(C\right)$ then -

All the face cards from a pack of $52$ playing cards are removed. From the remaining pack half of the cards are randomly removed without looking at them and then randomly drawn two cards simultaneously from the remaining. If the probability that, two cards drawn are both aces, is $\frac{p\left({}^{38}C_{20}\right)}{{}^{40}C_{20}.{}^{20}C_2}$, then find the value of $p$.

In a knockout tournament $2^n$ equally skilled players; $S_1, S_2........S_{2^n}$ are participating. In each round players are divide in pair at random and winner from each pair moves in the next round. If $S_2$ reaches the semifinal then the probability that $S_1$ wins the tournament is $\frac{1}{20}$. Find the value of '$n$'.

There is a three volume dictionary among $40$ books arranged on a shelf in random order. Then the reciprocal of probability of these volumes standing in increasing order from left to right (the volumes are not necessarily kept side by side) is

A party of $23$ persons take their seats at a round table. The odds against two persons sitting together are

The probability that $4^{\text{th }}$power of a positive integer ends in the digit $\lambda$ is $P\left(\lambda \right)$
 

If $4$ whole numbers taken at random are multiplied together, then

A mapping is selected at random from all the mappings defined on the set a consisting of three distinct elements. Probability that the mapping selected is one to one is $\frac{p}{q}$ (where $p$ and $q$ are co-prime natural numbers) then $p+q$ is

Three numbers are chosen at random without replacement from $\{1,2,3,\dots \dots ,10\}.$ The probability that the minimum of the chosen numbers is $3$ or their maximum is $7$ is $\frac{p}{q}$ then $(q-2p)$ equals. (Where $p \& q$ are co-prime natural numbers)

Seven digits from the number $1, 2, 3, 4, 5, 6, 7, 8 \& 9$ are written in random order. The probability that this seven-digit number is divisible by $9$ is $\frac{p}{q}$ then $(p+q)$ equals. Where $p \& q$ are co-prime natural numbers)

A boy has a collection of blue and green marbles. The number of blue marbles belong to the set $\{2, 3, 4, \dots ,13\}$. If two marbles are chosen simultaneously, and at random from his collection, then the probability that they have different colours is $\frac{1}{2}.$ Possible number of blue marbles 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

A player tosses a coin and scores $1 \& 2$ points for every head and tail repectively that turns up. If he has to play on until his score reaches or passes $n$. If $P_n\forall n\ge 1$ denotes the probability of getting a score of exactly $n$, then the value of $P_n$ is

Hat $\mathrm{W}$ contains two white balls and a black ball. Hat $\mathrm{B}$ contains two black balls and one white ball. At random, one of the following strategies is selected:

Two different balls are drawn from hat $\mathrm{W}$.

Two different balls are drawn from hat $\mathrm{B}$. 

One ball is drawn from each hat.

The probability of getting at least one white ball will be :

Three critics review a book. Odds in favour of the book are $5:2, 4:3 \& 3:4$ respectively for the three critics. The probability that majority are in favour of the book is