Question

The probability of getting a number between 1 and 100 which is only divisible by 1 and itself is:
A. \[\dfrac{1}{4}\]
B. \[\dfrac{1}{2}\]
C. \[\dfrac{{25}}{{98}}\]
D. \[\dfrac{1}{3}\]

Answer 1

Hint: In this particular problem find the total number between 1 and 100 that are prime numbers and then find the probability of them by using the formula \[{\text{Probability of favourable outcome}} = \dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total number of outcomes}}}}\].

Complete step by step answer:
As we know that we are asked the probability of numbers which are divisible by 1 and itself only.
So as per the definition of prime and composite numbers prime numbers are those numbers which are only divisible by 1 and the number itself like 2, 3, 5 etc. While the composite numbers are those which are divisible by 1, number itself and any other number also like 4, 6, 8 etc.
So, we had to find the probability of getting a prime number between 1 and 100.
Now as we know that the according to the probability formula probability of getting a favourable outcome \[ = \dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total number of outcomes}}}}\].
So here favourable outcome is prime number between 1 and 100.
As we know that there are a total of 98 numbers between 1 and 100 i.e. {2, 3, 4, 5, 6, …….. 95, 96, 97, 98, 99}.
So, now as per the above definition of prime numbers, the prime numbers between 1 and 100 will be {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}.
So, there are a total of 25 prime numbers between 1 and 100.
And there are 98 total numbers between 1 and 100.
So, probability of getting a prime number between 1 and 100 = \[\dfrac{{{\text{Number of prime numbers between 1 and 100}}}}{{{\text{Total number of numbers between 1 and 100}}}}\]
Probability of getting a prime number between 1 and 100 = \[\dfrac{{25}}{{{\text{98}}}}\]

Hence, the correct option will be C.

Note: Whenever we face such types of problems then first, we have to find the type of number whose probability is asked. And recall the definition of prime and composite numbers. And after that we can count the total prime numbers in the specified range. And then apply the probability formula that is probability of getting a prime number between 1 and 100 = \[\dfrac{{{\text{Number of prime numbers between 1 and 100}}}}{{{\text{Total number of numbers between 1 and 100}}}}\] to get the required answer.