Bayes' Theorem

All the jacks, queens, kings and aces of a regular $52$ cards deck are taken out. The $16$ cards are thoroughly shuffled and my opponent, a person who always tells the truth, simultaneously draws two cards at random and says, 'I hold at least one ace'. The probability that he holds two aces is

Let $A$ and $B$ are events of an experiment and $P\left(A\right)=\frac{1}{4}, P\left(A\cup B\right)=\frac{1}{2}$, then value of $P\left(\frac{B}{A^C}\right)$ is

If odds against solving a question by three students are $2:1, 5:2$ and $5:3$, respectively, then probability that the question is solved only by one student is

A bag contains $n$ white and $n$ black balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each pair consists of one white and one black ball, is

A hat contains a number of cards with $30\%$ white on both sides, $50\%$ black on one side and white on the other side, $20\%$ black on both sides. The cards are mixed up and a single card is drawn at random and placed on the table. Its upper side shows up black. The probability that its other side is also black is
 

Let $A$ and $B$ be two events such that $P\left(\overline{A\cup B}\right)=\frac{1}{6}, P\left(A\cap B\right)=\frac{1}{4}$, and $P\left(\overline{A}\right)=\frac{1}{4}$, where $\overline{A}$ stands for complement of event $A$. Then events $A$ and $B$ are

Two dice are rolled one after the other. The probability that the number on the first is smaller than the number on the second, is

If $A$ and $B$ are two independent events such that $P\left(A\right)=\frac{1}{2}, P\left(B\right)=\frac{1}{5}$, then

An artillery target may be either at point $\mathrm{I}$ with probability $\frac{8}{9}$ or at point $\mathrm{II}$ with probability $\frac{1}{9}$. We have $55$ shells, each of which can be fired either at point $\mathrm{I}$ or $\mathrm{II}$. Each shell may hit the target, independent of the other shells, with probability $\frac{1}{2}$. Maximum number of shells must be fired at point $\mathrm{I}$ to have maximum probability is,

If $A$ and $B$ are two events such that $P\left(A\right)=0.6$and $P\left(B\right)=0.8,$ if the greatest value that $P\left(\frac{A}{B}\right)$ can have is $p,$ then the value of $8p$ is,

Thirty-two players ranked $1$ to $32$ are playing in a knockout tournament. Assume that in every match between any two players, the better ranked player wins. The probability that ranked $1$ and ranked $2$ players are winner and runner up respectively is $p,$ then the value of $\left[\frac{2}{p}\right]$ is, where $\left[.\right]$ represents the greatest integer function,

Suppose $A$ and $B$ are two events with $P\left(A\right)=0.5$ and $P\left(A \cup B\right)=0.8.$ Let $P\left(B\right)=p$ if $A$ and $B$ are mutually exclusive and $P\left(B\right)=q$ if $A$ and $B$ are independent events, then the value of $\frac{q}{p}$ is,

An urn contains three red balls and $n$ white balls. Mr. $A$ draws two balls together from the urn. The probability that they have the same colour is $\frac{1}{2}$.Mr. $B$ draws one balls from the urn, notes its colour and replaces it. He then draws a second ball from the urn and finds that both balls have the same colour is $\frac{5}{8}$. The possible value of $n$ is______________.

An unbiased coin is tossed $6$ times. The probability that third head appears on the sixth trial is

If A and B are two independent events such that $P\left(A\right)=\frac{1}{2}, P\left(B\right)=\frac{1}{5}$, then

If two loaded dice each have the property that $2$ or $4$ is three times as likely to appear as $1, 3, 5,$ or $6$ on each roll. When two such dice are rolled, the probability of obtaining a total of $7$ is $p$, then the value of $\left[\frac{1}{p}\right]$ is, where $\left[x\right]$ represents the greatest integer less than or equal to $x.$

If any four numbers are selected and they are multiplied, then the probability that the last digit will be $1, 3, 5$ or $7$ is

A box contains $2$ black, $4$ white, and $3$ red balls. One ball is drawn at random from the box and kept aside. From the remaining balls in the box, another ball is drawn at random and kept aside the first. This process is repeated till all the balls are drawn from the box. The probability that the balls drawn are in the sequence of $2$ black, $4$ white, and $3$ red is

Let $A$ and $B$ two events. Suppose $P\left(A\right)=0.4, P\left(B\right)= p,$and $P\left(A\cup B\right)=0.7$. The value of $p$ for which $A$ and $B$ are independent is

Events $A$ and $C$ are independent. If the probabilities relating $A, B$ and $C$ are $P\left(A\right)=\frac{1}{5}, P\left(B\right)=\frac{1}{6}; P\left(A\cap C\right)=\frac{1}{20}; P\left(B\cup C\right)=\frac{3}{8}$. Then,