A company has two plants $P_1$ and $P_2$ which produces 60% and 40% of pens respectively. At plant $P_1$ 80% of pens are rated of grade A and at plant $P_2$ 90% of pens are rated of grade A. A pen is picked up at random and is found to be of grade A. The probability that it has manufactured from plant $P_1$ is:

The probability that in a year of the 22nd century choosen at random, there will be 53 Sundays is:

A random variable X has the following probability distribution:

The variance of the X will be:

A die is thrown thrice. If the first throw is a four, then the chance of getting 11 as total sum, is

Consider an experiment of tossing 3 coins simultaneously. Define the following events:

$E$ = [Three heads or three tails appear]

$F$ = [At least two heads appear] and

$G$ = [At most two heads appear]

Choose the correct option:

A bag contains 5 white, 4 black and 3 red balls. Which of the following statements are True?

A. The probability of drawing three red balls one by one without replacement is $\frac{1}{220}$

B. One by one three balls drawn without replacement then probability that third ball is red is $\frac{1}{4}$

C. The probability of drawing 2 white balls one by one without replacement is $\frac{3}{4}$

D. One by one 3 balls are drawn without replacement then probability that third ball is red is $\frac{3}{4}$

Choose the correct answer from the options given below:

It is given that only 0.1% of a large population have COVID infection. In this population, the reliability of COVID RTPCR-test is specified as follows :

For persons having COVID, 90% of the test detects the disease but 10% goes undetected. For persons not having COVID, 99% of the test is judged COVID negative but 1% are diagnosed as COVID positive.

Based on the above informations, answer the question :

The probability of the person tested as COVID positive, given that he is actually having COVID is :