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

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.

Define conditional probability and state the multiplication theorem.

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 A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], find AB and BA and state whether they are equal.