Bag $I$ contains $3$ red and $4$ black balls. Bag $II$ contains $5$ red and $6$ black balls. If one ball is drawn at random from one of the bags, and it is found to be red, then the probability that it was drawn from Bag $II$, is

For $k=1, 2, 3$,the box $B_k$ contains $k$ red balls and $(k+1)$ white balls. Let $P(B_1) = \frac{1}{2}$,$P(B_2) = \frac{1}{3}$,and $P(B_3) = \frac{1}{6}$. $A$ box is selected at random and a ball is drawn from it. If a red ball is drawn,then the probability that it comes from box $B_2$ 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

$A$ company produces $10,000$ items per day. On a particular day, $2500$ items were produced on machine $A$, $3500$ on machine $B$, and $4000$ on machine $C$. The probability that an item produced by machines $A, B, C$ is defective is $2 \%$, $3 \%$, and $5 \%$, respectively. If one item is selected at random from the output and is found to be defective, then the probability that it was produced by machine $C$ is:

In four schools $B_1, B_2, B_3, B_4$,the percentage of girl students is $12, 20, 13, 17$ respectively. From a school selected at random,one student is picked up at random and it is found that the student is a girl. The probability that the school selected is $B_2$ is:

Box $I$ contains $30$ cards numbered $1$ to $30$ and Box $II$ contains $20$ cards numbered $31$ to $50$. $A$ box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box $I$ is