Probability

If A and B are any two events having P(A ∪ B) = ¹/₂ and P(Ā) = ²/₃, then the probability of Ā ∩ B is?

Probability that a truck stopped at a roadblock will have faulty brakes or badly worn tires are 0.23 and 0.24 respectively. Also, the probability is 0.38 that a truck stopped at the roadblock will have faulty brakes and/or badly worn tires. Find the probability that a truck stopped at this roadblock will have faulty brakes as well as badly worn tires.

If a person visits his dentist, suppose the probability that he will have his teeth cleaned is 0.48, the probability that he will have a cavity filled is 0.25, the probability that he will have a tooth extracted is 0.20, the probability that he will have a teeth cleaned and a cavity filled is 0.09, the probability that he will have his teeth cleaned and a tooth extracted is 0.12, the probability that he will have a cavity filled and a tooth extracted is 0.07, and the probability that he will have his teeth cleaned, a cavity filled, and a tooth extracted is 0.03. What is the probability that a person visiting his dentist will have at least one of these things done to him?

Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that 

(a) you both enter the same section? 

(b) You both enter the different sections?

Three letters are dictated to three persons and an envelope is addressed to each of them, the letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.

Three squares of chess board are selected at random. Find the probability of getting 2 squares of one colour and other of a different colour.

If 4-digit numbers greater than 5,000 are randomly formed from the digits0, 1, 3, 5, and 7, what is the probability of forming a number divisible by 5 when, 

(i) the digits are repeated? 

(ii) the repetition of digits is not allowed?