$A$ 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

$A$ factory has two machines $A$ and $B$. Past record shows that machine $A$ produced $60 \%$ of the items of output and machine $B$ produced $40 \%$ of the items. Further,$2 \%$ of the items produced by machine $A$ and $1 \%$ produced by machine $B$ were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine $B$?

$A$ laboratory blood test is $99 \%$ effective in detecting a certain disease when it is in fact,present. However,the test also yields a false positive result for $0.5 \%$ of the healthy persons tested (that is,if a healthy person is tested,then,with probability $0.005,$ the test will imply he has the disease). If $0.1 \%$ of the population actually has the disease,what is the probability that a person has the disease given that his test result is positive?

Suppose that a bag $A$ contains $n$ red and $2$ black balls and another bag $B$ contains $2$ red and $n$ black balls. One of the two bags is selected at random and two balls are drawn from it at a time. When it is known that the two balls drawn are red,if the probability that those two balls drawn are from bag $A$ is $\frac{6}{7}$,then $n=$

In a test, a student either guesses, copies, or knows the answer to a multiple-choice question with four choices having one correct answer. The probability that he guesses the answer is $\frac{1}{3}$ and the probability that he copies it is $\frac{1}{12}$. The probability that his answer is correct given that he copied it is $\frac{1}{6}$. The probability that he knew the answer, given that he has correctly answered it, is

The probability of men getting a certain disease is $\frac{1}{2}$ and that of women getting the same disease is $\frac{1}{5}$. The blood test that identifies the disease gives the correct result with probability $\frac{4}{5}$. Suppose a person is chosen at random from a group of $30$ males and $20$ females,and the blood test of that person is found to be positive. What is the probability that the chosen person is a man?