Let $A, B,$ and $C$ be three mutually independent events. Consider the two statements $S_1$ and $S_2$:
$S_1 : A$ and $B \cup C$ are independent.
$S_2 : A$ and $B \cap C$ are independent.
Then:

Assume that the probability of a patient having a heart attack is $40 \%$. It is also assumed that a meditation and yoga course reduces the risk of a heart attack by $30 \%$ and the prescription of a certain drug reduces its risk by $25 \%$. $A$ patient can choose any one of the two options with equal probability. Given that a patient selected at random suffers a heart attack after choosing one of the two options,find the probability that the patient followed the course of meditation and yoga.

$A$ bag contains $6$ balls. If $4$ balls are drawn at a time and all of them are found to be red,then the probability that exactly $5$ of the balls in the bag are red is

In a toy factory,machines $A, B$,and $C$ are used to manufacture $30 \%, 40 \%$,and $30 \%$ of the output,respectively. The probabilities of toys made by machines $A, B$,and $C$ being defective are $2 \%, 3 \%$,and $1 \%$,respectively. $A$ toy is taken from the factory and is found to be defective. The probability that it was manufactured by machine $B$ 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:

$A$ man is known to speak truth $3$ out of $4$ times. He throws a die and reports that it is a six. Find the probability that it is actually a six.