There are two coins, one unbiased with probability $\frac{1}{2}$ of getting heads and the other one is biased with probability $\frac{3}{4}$ of getting heads. $A$ coin is selected at random and tossed. It shows heads up. Then, the probability that the unbiased coin was selected is

$A$ 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?

There are three coins. One is a two-headed coin (having heads on both faces),another is a biased coin that comes up heads $75 \%$ of the time,and the third is an unbiased coin. One of the three coins is chosen at random and tossed; it shows heads. What is the probability that it was the two-headed coin?

An insurance company insured $2000$ scooter drivers,$4000$ car drivers,and $6000$ truck drivers. The probabilities of an accident are $0.01, 0.03,$ and $0.15$ respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

$A$ bag contains $4$ red and $4$ black balls,another bag contains $2$ red and $6$ black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

Box $A$ contains $2$ black and $3$ red balls,while Box $B$ contains $3$ black and $4$ red balls. Out of these two boxes,one is selected at random; and the probability of choosing Box $A$ is double that of Box $B$. If a red ball is drawn from the selected box,then the probability that it has come from Box $B$ is: