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.

Box $A$ contains $2$ black and $3$ red balls,while Box $B$ contains $3$ black and $4$ red balls. Out of these two boxes,one is selected at random; and the probability of choosing Box $A$ is double that of Box $B$. If a red ball is drawn from the selected box,then the probability that it has come from Box $B$ is:

$A$ pack of playing cards was found to contain only $51$ cards. If the first $13$ cards which are examined are all red,then the probability that the missing card is black,is

$A$ 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 $.....$ .

There are $10$ coins in a box,out of which $8$ are normal and the remaining are with heads on both sides. $A$ coin is chosen at random from the box and tossed $6$ times. If it shows heads each time,then the probability that the selected coin has heads on both sides is