Question

Three numbers are chosen from 1 to 30 randomly. the probability that they are not consecutive is:

Answer 1

Let us find the sample space S first.

The given digits are 1,2,3,4,…….30

We have to choose 3 numbers out of 30. This can be done in 30C3 = 30(29)(28)/3.2.1 = 4060

Therefore n(S) = 30C3

The desired event E is that the 3 numbers so chosen must consecutive.

So, E = { (1,2,3), (2,3,4), (3,4,5),……,,,,,,,,,,,,(28,29,30)}. Obviously, there are 28 triplets.

Hence, n(E) = 28

Hence the probability of 3 consecutive numbers from 1 to 30 is = n (E)/n(S) = = 28/4060 = 1/145.

Answer 2

If the three numbers have to be consecutive:

Then the favorable outcomes would be

(1,2,3),(2,3,4).....(27,28,29),(28,29,30) i.e. 28 outcomes.

Now the total no. of outcomes would be:

We have to select 3 nos. out of 30 nos. irrespective of the order of the numbers.

Therefore, the total number of outcomes would be 30C3.

The probability would be 28/30C3 which will give us 1/145.

The answer will be 1/145.