Question

: An integer is chosen at random from 1 to 50. Find the probability that the number is
A. Divisible by 5
B. A perfect cube
C. A prime number

Answer 1

Hint: We had to find the probability of choosing the integer which is divisible by 5, a perfect cube and a prime number, for all we will go with the basic formula of probability which says that probability of favourable outcome$=\dfrac{\text{Number }~\text{ of }~\text{ favourable }~\text{ outcome}}{\text{Total }~\text{number }~\text{of outcome }~}$

Complete step by step answer:
Moving ahead with the question in step to step method;
Total number of outcomes$=50$
So for probability of choosing an integer which is divisible by 5 ranging from 1 to 50, so first we have to count the total number of integer that are divisible by 5;
Number divisible by 5 (from 1 to 50)$=5,10,15,20,25,30,35,40,45,50$i.e. total 10 number of integers are there that are divisible by 5. So for the probability of choosing an integer which is divisible by 5 ranging from 1 to 50$=\dfrac{\text{Number }\!\!~\!\!\text{ of }\!\!~\!\!\text{ favourable }\!\!~\!\!\text{ outcome}}{\text{Total}~\text{number}~\text{of outcome}~}$
Number of favourable outcome$=10$
Total number of outcomes$=50$
So probability for choosing an integer which is divisible by 5$\begin{align}
  & =\dfrac{\text{Number }~\text{ of }~\text{ favourable }~\text{ outcome}}{\text{Total }~\text{number }~\text{of outcome }~} \\
 & =\dfrac{10}{50}=\dfrac{1}{5} \\
\end{align}$
In the same way, find the probability that the chosen number is cube root, so first look out the total cube roots ranging from 1 to 50 that are$1,8,27$that are three cube roots are there. So probability of choosing an integer which is cube root ranging from 1 to 50$\begin{align}
  & =\dfrac{\text{Number }~\text{ of }~\text{ favourable }~\text{ outcome}}{\text{Total }~\text{number }~\text{of outcome }~} \\
 & =\dfrac{3}{50} \\
\end{align}$
Similarly for finding the probability of choosing an integer which is prime number, so look out for the total number of prime number exist ranging from 1 to 50 that are$1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,$. So total prime numbers are 16. So for probability of choosing an integer which is a prime number ranging from 1 to 50$\begin{align}
  & =\dfrac{\text{Number }~\text{ of }~\text{ favourable }~\text{ outcome}}{\text{Total }~\text{number }~\text{of outcome }~} \\
 & =\dfrac{16}{50}=\dfrac{8}{25} \\
\end{align}$
Hence$\dfrac{1}{5},\dfrac{3}{50},\dfrac{8}{25}$ are the probability of choosing an integer which are divisible by 5, which are cube root and which are prime numbers.

Note: Numbers whose cube root is an integer is called cube root. So for finding the probability we will simply find out all possible outcomes and divide them by the total number of outcomes.