From a lot of 20 bulbs which include 5 defectives, a sample of 3 bulbs is drawn at random, one by one with replacement. Find the probability distribution of the number of defective bulbs. Also, find the mean of the distribution.

Find the shortest distance between the lines and .

Two groups are competing for the positions of the Board of Directors of a corporation. The probabilities that the first and second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.

Show that the relation R on the set Z of all integers defined by (x, y) ∈ R ⇔ (x – y) is divisible by 3 is an equivalence relation.

OR

A binary operation on the set A = {0, 1, 2, 3, 4, 5} is defined as

.

Write the operation table for a * b in A.

Show that zero is the identity for this operation * and each element ‘a’ ≠ 0 of the set is invertible with 6 – a, being the inverse of ‘a’.

Given,

OR

Find the inverse of the matrix , by using elementary row transformations.