$A$ bag contains $6$ balls. If $4$ balls are drawn at a time and all of them are found to be red,then the probability that exactly $5$ of the balls in the bag are red 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 ...... .

An urn contains $7$ red, $5$ white and $3$ black balls. Three balls are drawn randomly one after the other without replacement. If it is known that the first ball drawn is red and the second ball drawn is white, then the probability that the third ball drawn is not red is

$A$ bag contains $2n+1$ coins. It is known that $n$ of these coins have heads on both sides,whereas the other $n+1$ coins are fair. One coin is selected at random and tossed. If the probability that the toss results in heads is $\frac{31}{42}$,then the value of $n$ is

Meera visits only one of the two temples $A$ and $B$ in her locality. The probability that she visits temple $A$ is $\frac{2}{5}$. If she visits temple $A$,the probability that she meets her friend is $\frac{1}{3}$,whereas it is $\frac{2}{7}$ if she visits temple $B$. Meera met her friend at one of the two temples. The probability that she met her at temple $B$ is

Events $E_{1}$ and $E_{2}$ form a partition of the sample space $S$. $A$ is any event such that $P(E_{1}) = P(E_{2}) = \frac{1}{2}$,$P(E_{2} | A) = \frac{1}{2}$,and $P(A | E_{2}) = \frac{2}{3}$. Then $P(E_{1} | A)$ is: