A factory has two machines. Machine A makes 60% of the output with 2% defective, and machine B makes 40% with 5% defective. An item chosen at random is defective. Find the probability that it came from machine A.

If P(A) = 1/3 and P(B) = 1/2 and A and B are independent, find P(A ∩ B) and P(A ∪ B).

For a binomial distribution with n = 10 and p = 0.4, find the mean and the variance.

A fair coin is tossed 5 times. Find the probability of getting exactly 3 heads.

A bag has 4 red and 6 black balls. Two are drawn one after another without replacement. Find the probability that both are red.

Define conditional probability and state the multiplication theorem.

Maximise Z = 3x + 4y subject to x + y ≤ 4, x ≥ 0 and y ≥ 0.