Question

Prime numbers between \[1\] to \[20\] .What is the probability of getting an odd number?

Answer 1

Hint: In order to solve this question, first of all we list down all the prime numbers from \[1\] to \[20\] . After that we will find out what are the favourable outcomes and total outcomes for the event of getting an odd number and then finally use the formula of finding a probability of event i.e., \[P\left( E \right) = \dfrac{{Number{\text{ }}of{\text{ }}favourable{\text{ }}outcomes}}{{Number{\text{ }}of{\text{ }}total{\text{ }}outcomes}}\] and hence we get the required result.

Complete step-by-step answer:
First of all, let’s recall the definition of prime numbers.
A number whose factors are \[1\] and itself only is known as a prime number.
So, now lists down all the prime numbers from \[1\] to \[20\] .
Therefore, we get \[2,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}9,{\text{ }}11,{\text{ }}13,{\text{ }}17,{\text{ }}19\]
Now, we have to find out the probability of getting an odd number.
We know that odd numbers are those which cannot be divided by \[2\]
So, now lists down all the odd numbers from the set of prime numbers between \[1\] to \[20\]
Therefore, we get
\[3,{\text{ }}5,{\text{ }}7,{\text{ }}9,{\text{ }}11,{\text{ }}13,{\text{ }}17,{\text{ }}19\]
Thus, favourable outcomes of getting an odd number equals to \[3,{\text{ }}5,{\text{ }}7,{\text{ }}9,{\text{ }}11,{\text{ }}13,{\text{ }}17,{\text{ }}19\]
Therefore, number of favourable outcomes of getting an odd number \[ = 8\]
And according to the question, number of total outcomes \[ = 9\]
Now by using the formula of finding a probability of event i.e., \[P\left( E \right) = \dfrac{{Number{\text{ }}of{\text{ }}favourable{\text{ }}outcomes}}{{Number{\text{ }}of{\text{ }}total{\text{ }}outcomes}}\]
we calculate the probability of getting an odd number
Therefore, we get
\[P\left( E \right) = \dfrac{{Number{\text{ }}of{\text{ }}favourable{\text{ }}outcomes}}{{Number{\text{ }}of{\text{ }}total{\text{ }}outcomes}}{\text{ }} = \dfrac{8}{{9}}\]
On simplifying, we get
\[P\left( E \right) = \dfrac{8}{{9}}\]
So, the correct answer is \[P\left( E \right) = \dfrac{8}{{9}}\]”.

Note: The common mistake students make while solving this question is one might confuse the definition of prime numbers with the definition of odd numbers. Also, some students make the mistake of considering all the odd numbers as prime numbers, but this is wrong. For example, numbers like \[9,{\text{ }}15\] are odd but not prime. So, this mistake must be avoided. And also remember that \[1\] is not considered as a prime number. Except 2 all prime numbers are odd.