Bag $A$ contains $3$ white and $7$ red balls,and bag $B$ contains $3$ white and $2$ red balls. One bag is selected at random and a ball is drawn from it. What is the probability that the ball was drawn from bag $A$,given that the ball drawn is white?

$A$ computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces $20 \%$ and plant $T_2$ produces $80 \%$ of the total computers produced. $7 \%$ of computers produced in the factory turn out to be defective. It is known that $P(\text{defective} | T_1) = 10 P(\text{defective} | T_2)$. $A$ computer produced in the factory is randomly selected and it is not defective. Then the probability that it is produced in plant $T_2$ is

$A$ and $B$ are two events of a random experiment such that $P(B)=0.4$,$P(A \cap \bar{B})=0.5$,and $P(A \cup B) + P\left(\frac{B}{A \cup \bar{B}}\right) = 1.15$. Then $P(A) = $

$A$ man speaks truth $2$ out of $3$ times. He picks one of the natural numbers in the set $S=\{1, 2, 3, 4, 5, 6, 7\}$ and reports that it is even. The probability that it is actually even 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?

$A$ person is known to speak the truth $3$ out of $4$ times. If that person picks a card at random from a pack of $52$ cards and reports that it is a king,then the probability that it is actually a king is