Two groups are competing for the position on the board of directors of a corporation. The probabilities that the first and the second groups will win are $0.6$ and $0.4$ respectively. Further,if the first group wins,the probability of introducing a new product is $0.7$ and the corresponding probability is $0.3$ if the second group wins. Find the probability that the new product was introduced by the second group. (in $/9$)

$A$ factory has a total of three manufacturing units,$M_1, M_2$,and $M_3$,which produce bulbs independently. The units $M_1, M_2$,and $M_3$ produce bulbs in the proportions of $2: 2: 1$,respectively. It is known that $20\%$ of the bulbs produced in the factory are defective. It is also known that,of all the bulbs produced by $M_1, 15\%$ are defective. Suppose that,if a randomly chosen bulb produced in the factory is found to be defective,the probability that it was produced by $M_2$ is $\frac{2}{5}$. If a bulb is chosen randomly from the bulbs produced by $M_3$,then the probability that it is defective is $.....$ .

In a school there are $3$ sections $A, B$ and $C$. Section $A$ contains $20$ girls and $30$ boys,section $B$ contains $40$ girls and $20$ boys and section $C$ contains $10$ girls and $30$ boys. The probabilities of selecting the section $A, B$ and $C$ are $0.2, 0.3$ and $0.5$ respectively. If a student selected at random from the school is a girl,then the probability that she belongs to section $A$ is

Four boxes $A, B, C$ and $D$ contain $5000, 3000, 2000$ and $1000$ fuses respectively. The percentages of defective fuses in these boxes are $3\%, 2\%, 1\%$ and $0.5\%$ respectively. If a fuse selected at random from one of the boxes is found to be defective, then the probability that it has come from box $D$ is

$A$ box $B_1$ contains $3$ blue balls and $6$ red balls. Another box $B_2$ contains $8$ blue balls and $n$ red balls $(n \in N)$. $A$ ball selected at random from a box is found to be red. If $p$ is the probability that this red ball drawn is from box $B_2$,then

Bag $A$ contains $6$ Green and $8$ Red balls and bag $B$ contains $9$ Green and $5$ Red balls. $A$ card is drawn at random from a well-shuffled pack of $52$ playing cards. If it is a spade,two balls are drawn at random from bag $A$,otherwise two balls are drawn at random from bag $B$. If the two balls drawn are found to be of the same colour,then the probability that they are drawn from bag $A$ is