Probability Tes...

The chance that Doctor A will diagonise disease X correctly is $$60\%$$. The chance that a patient will die by his treatment after correct diagnosis is $$40\%$$ and the chance of death after wrong diagnosis is $$70\%.$$ A patient of Doctor A who had disease X died. The probability that his disease was diagonised correctly is

A coin is tossed $$3$$ times. If $$E$$ denotes the event in which heads appear at least twice and $$F$$ denotes the event in which head comes first, then $$P(\displaystyle {E}|{F})=$$

An urn $$A$$ contains $$2$$ white and $$3$$ black balls. Another urn $$B$$ contains $$3$$ white and $$4$$ black balls.Out of these two urns, one is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that
I. It is from urn $$A$$ is $$21/40$$,
II. It is from urn $$B$$ is $$20/41$$
Which of the following statements is correct

If $$A$$ and $$B$$ are independent events such that $$P\left( A \right) >0,P\left( B \right) >0$$, then

If $$ \bar E$$ and $$ \bar F$$ are the complementary events of events $$E$$ and $$F$$, respectively, and if $$0 < P(F)<1$$, then

                          
For any two events A and B, the conditional probability $$P\left ( B/A \right )=\frac{P\, \left ( B \,\cap\,A \right )}{P\, \left ( A \right )}$$ and ifAand B are independent
$$P\left ( B \cap A \right )=P\left ( B \right ).P\left ( A \right )$$  So, $$P\left ( B/A \right )=P\left ( B \right )$$

A lot contains 50 defective and 50 non-defective bulbs. Two bulbs are drawn at random one at a time with replacement. The events A, B, C are defined as:
A : 1st bulb is defective
B : 2nd bulb is non-defective
C : both are defective or both are non-defective
then,

If 6 persons are selected, the probability that there will be two trio's in which exactly one trio is of the same family is

In a locality, out of $$5000$$ people residing, $$1200$$ are above $$30$$ years of age and $$3000$$ are females. Out of the $$1200$$ who are above $$30$$, two hundred are females. Suppose, after a person is chosen you are told that the person is a female. What is the probability that she is above $$30$$ years of age?

Four different objects $$1,2,3,4$$ are distributed at random in four places marked $$1, 2, 3, 4$$. What is the probability that none of the objects occupy the place corresponding to their number?

There are 18 points in a plane such that no three of them are in the same line except five points which are collinear. The number of triangles formed by these points is