Question

An integer is chosen at random between $1$ and $100$. The probability that it is divisible by $8$ is $\frac{m}{25}$. Find $m$

Answer 1

Number of integers between $1$ and $100$ are $100$.

Clearly, this is the size of the sample space also.

The first integer between $1$ and $100$ divisible by $8$ is $8$ and the last integer in the same range divisible by $8$ is $96$.

So, these numbers form an A.P. with first term $8$ and common difference $8$.

So, we have $a_n=96,a_1=d=8\Rightarrow n-1=\frac{a_n-a_1}{d}=\frac{96-8}{8}=11\Rightarrow n=11+1=12$.

So there are $12$ integers between $1$ and $100$ that are divisible by $8$ and each number is equally likely of being picked.

So the probability that a randomly chosen integer in the given range is divisible by $8$ is $\frac{12}{100}=\frac{3}{25}$.

hence $m=3$