Question

There are 25 stamps numbered from 1 to 25 in a box. If a stamp is drawn at random from the box. The probability that the number on the stamp drawn is a prime number is
A. $\dfrac{{12}}{{25}}$
B. $\dfrac{{13}}{{25}}$
C. $\dfrac{9}{{25}}$
D. $\dfrac{6}{{25}}$

Answer 1

Hint: At first we’ll find the number of ways of selection any 1 out of 25 stamps, then we’ll find the numbers of prime numbers from 1 to 25 and the number of ways of selecting one prime number out of all prime number. Then using the formula for probability, i.e., probability of an event \[ = \dfrac{{favourable{\text{ outcomes}}}}{{total{\text{ possible outcomes}}}}\], we’ll find the required probability.

Complete step by step Answer:

Given data: Total number of stamps in the box=25
We know that the number of ways of selecting any ‘r’ elements out of a total ‘n’ number of elements irrespective of the order of ‘r’ elements is given by ${}^n{C_r}$,
Where, ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$and $n! = n(n - 1)(n - 2)(n - 3)(n - 4)..........(3)(2)(1)$
Therefore the number of ways that any 1 stamp is chosen out of 25$ = {}^{25}{C_1}$
Prime numbers from 1 to 25 are 2, 3, 5, 7, 11, 13, 17, 19 and 23
Therefore, the number of prime numbers between 1 and 25=9
The number of ways that any prime numbered stamp is chosen out of 9 primes$ = {}^9{C_1}$
As, probablilty of an event \[ = \dfrac{{favourable{\text{ outcomes}}}}{{total{\text{ possible outcomes}}}}\]
Therefore, the probability that the number on the stamp drawn is a prime number$ = \dfrac{{{}^9{C_1}}}{{{}^{25}{C_1}}}$
Using ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$, we get,
$ = \dfrac{{\dfrac{{9!}}{{1!\left( {9 - 1} \right)!}}}}{{\dfrac{{25!}}{{1!\left( {25 - 1} \right)!}}}}$
Now using \[n! = n\left( {n - 1} \right)!\] , we get,
$ = \dfrac{{\dfrac{{9 \times 8!}}{{1!\left( 8 \right)!}}}}{{\dfrac{{25 \times 24!}}{{1!\left( {24} \right)!}}}}$
On cancelling common terms we get,
$ = \dfrac{9}{{25}}$
Hence, the probability that the number on the stamp drawn is a prime number is $\dfrac{9}{{25}}$

Note: Here we have taken prime numbers like 2, 3, 5 and, 7, but some students may count 1 as well as it satisfies both the condition according to the definition of a prime number which is, A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. But there is also a definition of prime number that has only two factors which are not satisfied by 1, hence it is not included under the category of prime numbers.