MATHEMATICS - OBJECTIVE -3

11. If three distinct numbers are chosen randomly from the first 100 natural numbers, then the probability that all three of them are divisible by both 2 and 3 

(a) 4/55                 (b) 4/35

(c) 4/33                 (d) 4/1155

 Answer (d)

Solution method: A distinct number that is randomly chosen from a population of big size with a small fixed sample lot is called as Bayer's approximation of probability ratios. This is represented by

          nCr=  n! / r! ( n - r)!

There are 16 digits divisible by 6 ( common factor of 2 and 3 = 2x3 =6) So r=16, N=100

Number of ways in which 3 numbers are selected from these 16 numbers is  16C3

Number of ways in which 3 numbers are selected randomly from 100 numbers is 100C3 

 nCr = ( 16 x 15 x 14 / 3 x 2 x 1 )

         _______________________   = 4 / 1155

         ( 100 x 99 x 98 / 3 x 2 x 1 )

Explanation: This type of analysis is done on a huge set of numbers ,within which ,a smaller subset is selected for probability analysis. This can be considered as the Bayer theorem corollary,since probability theory can be used to negotiate plans for immediate future .This is used in market speculation in current world. As shown above, a definite answer to probability of numbers indicates futuristic ways on market trading in stocks and business prediction in today's competitive world. So here ,it shows that probability is indeed a science of fairly good amount of accuracy .

Chevalier De Mere's problem ( a classic case)shows that Bayer's theory is not fully accurate. This needs approximations as per the above case.