Probability Tes...

$$A, B$$ and $$C$$ are three mutually exclusive and exhaustive events such that $$P(A) = 2 P(B) = 3P(C)$$. What is $$P(B)$$?

Consider the following in respect of two events $$A$$ and $$B$$:
(1) $$P(A$$ occurs but not $$B)=P(A)-P(B)$$ if $$B\subset A$$
(2) $$P(A$$ alone or $$B$$ alone occurs) $$=P(A)+P(B)-P(A\cap B)$$
(3) $$P(A\cup B)=P(A)+P(B)$$ if $$A$$ and $$B$$ are mutually exclusive
Which of the above is/are correct?

Consider the following statements:
(1) $$P(\bar { A } \cup B)=P(\bar { A } )+P(B)-P(\bar { A } \cap B)$$
(2) $$P(A\cap \bar B) = P(\bar B)-P(A\cap B)$$
(3) $$P(A\cap B)=P(B)P(A|B)\quad $$
Which of the above statements are correct?

For two events $$A$$ and $$B$$, let $$P(A) =\dfrac {1}{2}, P(A\cup B) = \dfrac {2}{3}$$ and $$P(A\cap B) = \dfrac {1}{6}$$, What is $$P(\overline {A} \cap B)$$ equal to?

If $$P(B) = \dfrac {3}{4}, P(A\cap B\cap \overline {C}) = \dfrac {1}{3}$$ and $$P(\overline {A}\cap B\cap \overline {C}) = \dfrac {1}{3}$$, then what is $$P(B\cap C)$$ equal to?

A survey of $$850$$ students in a University yields that $$680$$ students like music and $$215$$ like dance. What is the least number of students who like both music and dance?

In a class, $$54$$ students are good in Hindi only, $$63$$ students are good in Mathematics only and $$41$$ students are good in English only. There are $$18$$ students who are good in both Hindi and Mathematics. $$10$$ students are good in all three subjects. What is the number of students who are good in either Hindi or Mathematics but not in English?

If $$A$$ and $$B$$ are independent events, $$P(A)=0.1$$ and $$P(B)=0.9$$, then $$P\left( A\cup B \right) =$$ ____

In a survey of $$400$$ students in a school, $$100$$ are listed as taking apple juice, $$150$$ as taking orange juice and $$75$$ were listed as taking both apple as well as orange juice. Find how many students were taking "Neither apple nor orange juice"

Tickets numbered from $$1$$ to $$30$$ are mixed up and then a ticket is drawn at random. What is the probability that the drawn ticket has a number which is divisible by both $$2$$ and $$6$$?