AashishGautam wrote:
we have to find the probability that the number of numbers that will be divisible by 9. Wont we have to find the number of numbers to find the actual probability?
It's only sometimes true that you need to count all of the possibilities to answer a probability question. For example, if you were asked "if a random five-digit number is created using each of the digits 1, 3, 5, 7 and 9 once each, what is the probability the number will be odd?" there's no need to calculate how many five-digit numbers can be made. The result will be odd no matter what, so the probability is 1.
ThatDudeKnows wrote:
1/9 of positive integers are divisible by 9. Unless we insert some sort of bias to our set of possible 7-digit numbers that impacts divisibility by 9, our odds would be 1/9. The only digit we aren't allowed to use is 0. A number is divisible by 9 if the sum of the digits is divisible by 9. The absence of 0 from our universe does not create a bias, so we still have 1/9.
This justification of the answer isn't right, as you'll see if you try to apply the same reasoning to the same question, but where we're making 6-digit numbers instead of 7-digit numbers. The probability a random 6-digit number made from six of the digits from 1 through 9 is divisible by 9 is not 1/9 (it can't be, because we have 9C3 = 84 choices for the digits, and 84 is not a multiple of 9, so we can't get a denominator of 9). The answer to that question turns out to be 5/42 if my quick arithmetic is right.