What is the probability that a number selected at random from the first? 1,000 positive integers is? (exactly) divisible by 14 or 4??

 

According to the question,

LCM(14, 4) = 28

Let A be the event to select a number which are divisible by 14 from first 1000 positive integer.

B be the event to select a number which are divisible by 4 from first 1000 positive integer.

C be the event to select a number which are divisible by 28 from first 1000 positive integer.

Count of numbers divisible by 14 is

n(A) = [1000 / 14]

= [71.4285]

= 71

Count of numbers divisible by 4 is,

n(B) = [1000 / 4]

= 250

Count of numbers divisible by 28 is,

n(C) = [1000 / 28]

= [35.7142]

= 35

∴ Count of numbers divisible by 14 or 4 is,

= n(A) + n(B) - n(C)

= 71 + 250 - 35

= 286

∴ The probability for the number to be divisible by 14 or 4 is = 286 / 1000

= 0.286

∴ The probability that a number selected at random from the first 1000 positive integers which is divisible by 14 or 4 is 0.286