$A$ bag contains $5$ balls of unknown colours. There are equal chances that out of these five balls,there may be $0, 1, 2, 3, 4,$ or $5$ red balls. $A$ ball is taken out from the bag at random and is found to be red. The probability that it is the only red ball in the bag is

Two cards are drawn at random from a pack of $52$ playing cards. If both the cards drawn are found to be black in colour,then the probability that at least one of them is a face card is

$A$ bag contains $2n+1$ coins. It is known that $n$ of these coins have a head on both sides, whereas the remaining $n+1$ coins are fair. $A$ coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is $\frac{31}{42}$, then $n$ is equal to

In an entrance test,there are multiple-choice questions. There are four possible answers to each question,of which one is correct. The probability that a student knows the answer to a question is $9/10$. If he gets the correct answer to a question,then the probability that he was guessing is:

$A$ certain disease has a prevalence of $1\%$ in the population. $A$ diagnostic test for the disease has a sensitivity of $98\%$ and a specificity of $95\%$. If a person from this population tests positive, then the probability that they actually have the disease is... (in $\%$)

$A$ doctor is to visit a patient. From past experience,it is known that the probabilities that he will come by train,bus,scooter,or by other means of transport are respectively $\frac{3}{10}, \frac{1}{5}, \frac{1}{10},$ and $\frac{2}{5}$. The probabilities that he will be late are $\frac{1}{4}, \frac{1}{3},$ and $\frac{1}{12}$ if he comes by train,bus,and scooter respectively,but if he comes by other means of transport,then he will not be late. When he arrives,he is late. What is the probability that he comes by train?