Bag $I$ contains $3$ red,$4$ black,and $3$ white balls. Bag $II$ contains $2$ red,$5$ black,and $2$ white balls. One ball is transferred from Bag $I$ to Bag $II$,and then a ball is drawn from Bag $II$. The ball drawn is found to be black. What is the probability that the transferred ball was red?

$A$ bag contains $19$ unbiased coins and one coin with head on both sides. One coin is drawn at random and tossed,and a head turns up. If the probability that the drawn coin was unbiased is $\frac{m}{n}$,where $\operatorname{gcd}(m, n) = 1$,then $n^2 - m^2$ is equal to:

$A$ car manufacturing factory has two plants $X$ and $Y$. Plant $X$ manufactures $70 \%$ of cars and plant $Y$ manufactures $30 \%$ of cars. $80 \%$ of cars at plant $X$ and $90 \%$ of cars at plant $Y$ are rated as standard quality. $A$ car is chosen at random and is found to be standard quality. The probability that it has come from plant $X$ is

Suppose a girl throws a die. If she gets a $5$ or $6,$ she tosses a coin three times and notes the number of heads. If she gets $1, 2, 3$ or $4,$ she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head,what is the probability that she threw $1, 2, 3$ or $4$ with the die?

Three companies $C_1, C_2, C_3$ produce car tyres. $A$ car manufacturing company buys $40 \%$ of its requirement from $C_1, 35 \%$ from $C_2$ and $25 \%$ from $C_3$. The company knows that $2 \%$ of the tyres supplied by $C_1, 3 \%$ by $C_2$ and $4 \%$ by $C_3$ are defective. If a tyre chosen at random from the consignment received is found defective, then the probability that it was supplied by $C_2$ is:

An urn contains $6$ white and $9$ black balls. Two successive draws of $4$ balls are made without replacement. The probability,that the first draw gives all white balls and the second draw gives all black balls,is: