Question

The probability that a number selected at random from the numbers $1,2,3,...,99,100$ is a prime is
$A)0.4$
$B)0.25$
$C)0.45$
$D)0.43$

Answer 1

Hint: First, we will need to know about the prime numbers and probability concepts.
Prime numbers are the numbers that are divisible by themselves and $1$only or also known as the numbers whose factors are the given number itself.
Probability is the term mathematically with events that occur, which is the number of favorable events that divides the total number of the outcomes.
Formula used: $P = \dfrac{F}{T}$ where P is the probability, F is the possible favorable events and T is the total outcomes.

Complete step by step answer:
From the given that, there are total $100$ numbers in the set $1,2,3,...,99,100$.
From this, $1,2,3,...,99,100$ we have to check which prime number occurs.
We know that the factors of prime numbers are $1$ and themselves.
So we will first list all the numbers which have only two-factors $1$ and themselves.
The prime numbers between the numbers $1,2,3,...,99,100$ are $2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.$
In these numbers the factors are $1$ and themselves only, like take the number $73$ and try to divide with any of the numbers from $2,3,4,......$and the result is remainder non-zero.
We say the number is completely divisible only if the remainder is zero.
Hence, we get a total of $25$ prime numbers from the set of all numbers $1,2,3,...,99,100$
Thus we get the favorable event is $25$ and the total event is $100$(total count from the given numbers)
Therefore, the probability of the selected at random from the numbers $1,2,3,...,99,100$ is a prime is $P = \dfrac{F}{T} \Rightarrow \dfrac{{25}}{{100}}$
Now by the help of division operation, we get, $P = \dfrac{F}{T} \Rightarrow \dfrac{{25}}{{100}} \Rightarrow 0.25$

So, the correct answer is “Option B”.

Note: Every number greater than $1$ can be divided by at least one prime number.
$2$ is the only even prime number.
If the number is not a prime number then it is known as the composite number.
Composite numbers defined as the numbers having more than two factors ($1$, itself and any other numbers)