Question

The probability of choosing at random a number that is divisible by 6 or 8 from among 1 to 90 is equal to
A. $\dfrac{1}{6}$
B. $\dfrac{1}{30}$
C. $\dfrac{11}{80}$
D. $\dfrac{23}{90}$

Answer 1

Hint : Count the total numbers divisible by 6 or 8. Remember to subtract the counting of numbers which are divisible by both $6\,\,\text{or}\,\,8$ . Because otherwise they would be counted twice. After counting the total numbers divisible by 6 or 8, find the probability using the formula: $P(E)=\dfrac{\text{number}\,\,\text{of}\,\,\text{favourable}\,\,\text{outcomes}}{\text{total}\,\,\text{number}\,\,\text{of}\,\,\text{outcomes}}$

Complete step-by-step answer :
90 is a 15th number divisible by 6
There are 15 numbers divisible by 6 from 1 to 90
Total number divisible by 8:
The same way we have 90 divisible by 8 from 1 to 90 is the $\dfrac{88}{\text{8}}=$ 11th number divisible by 8
If we want to count numbers divisible by 6 or 8, we have to add up the numbers we calculated before and subtract the amount of numbers divisible by
$LCM(6,8)=24$ Because we have counted twice.
Total numbers divisible by 24: first find out how many numbers are divisible by 6 or 8 from 1 to 90
Total numbers divisible by 6:
We know that 90 is divisible by 6 and the next number which is divisible by 6 is 96 which is greater than 90.
So, ${{n}^{th}}$ number divisible by 6 is 90.
$6n=90$
$n=\dfrac{90}{6}$
$n=15$
We have that 72 is divisible by 24 and the next number divisible by 24 is 96 which is greater than 90.
So, the amount of numbers divisible by 6 or 8 from 1 to 90 is:
$15+11-3=23$
Now, probability of an event $P(E)$ is given by
$P(E)=\dfrac{\text{number}\,\,\text{of}\,\,\text{favourable}\,\,\text{outcomes}}{\text{total}\,\,\text{number}\,\,\text{of}\,\,\text{outcomes}}$
So, the probability of choosing a number divisible by 6 or 8 from 1 to 90.
$P\left( 6\,\,\text{or}\,\,8 \right)=\dfrac{\text{total numbers}\,\,\text{divisible by}\,\,6\,\,\text{or}\,\,8}{\text{total}\,\,\text{numbers}\,\,\text{chosen}\,\,}$
$P\left( 6\,\,\text{or}\,\,8 \right)=\dfrac{23}{90}$
So, the correct answer is “Option D”.

Note : The total numbers divisible by 6 or 8 from 1 to 90 can also be counted by the following method:
Total numbers divisible by 6 is $\dfrac{90}{6}=15$ and Total numbers divisible by 8 is $\dfrac{90}{8}=11.25$ (rounding to the smaller nearest integer). Similarly, we can find the total divisible numbers for any integer by this method.