Question

Cards marked with numbers $1,2,3,4,....20$ are well shuffled and a card is drawn at random. What is the probability that the number on the card is:
(i) a prime number?
(ii) divisible by 3?
(iii) a perfect square?

Answer 1

For the solution of part (i), find the total number of prime numbers there between $1$ to $20$ and then obtain the probability by using the probability formula. For part (ii), find the total number which are divisible by $3$ between $1$ to $20$ and then obtain the probability by using the probability formula. For the last part of the question, count the total number of perfect squares between $1$ to $20$ and then obtain the probability by using the probability formula. For the last part of the question.

Complete step-by-step answer:
(i)
If the number has two factors $1$ and the number itself, then the number is called a prime number.
From numbers $1$ to $20$ the prime numbers are $2,3,5,7,11,13,17,19$.
So, the total number of prime numbers from $1$ to $20$ are $8$.
The formula of the probability of an event is written as,
$P\left( A \right) = \dfrac{{{\text{Number of favourable outcome}}}}{{{\text{Total Number of favourable outcome}}}}$
Here, the number of favorable outcomes are $8$ and the total number of favorable outcomes are $20$.
So, the probability that the number on the card is prime is calculated as,
$P\left( A \right) = \dfrac{{\text{8}}}{{{\text{20}}}} \\\ = \dfrac{2}{5} \\\$
Therefore, the probability that the number on the card is a prime number is $\dfrac{2}{5}$.

(ii)
If the sum of all digits is a multiple of $3$ or divisibility by $3$, then the number will be divisible by $3$.
From numbers $1$ to $20$ the numbers divisible by $3$ are $3,6,9,12,15,18$.
So, the total numbers that are divisible by $3$ from $1$ to $20$ are $6$.
The formula of the probability of an event is written as,
$P\left( A \right) = \dfrac{{{\text{Number of favourable outcome}}}}{{{\text{Total Number of favourable outcome}}}}$
Here, the number of favorable outcomes are $6$ and the total number of favorable outcomes are $20$.
So, the probability that the number on the card is divisible by $3$ is calculated as,
$P\left( A \right) = \dfrac{{\text{6}}}{{{\text{20}}}} \\\ = \dfrac{3}{{10}} \\\$
Therefore, the probability that the number on the card is divisible by $3$ is $\dfrac{3}{{10}}$.

(iii)
If a number is made by squaring a whole number then it is called a perfect square.
From numbers $1$ to $20$ the perfect squares are $1,4,9,16$.
So, the total numbers that are perfect squares from $1$ to $20$ are $4$.
The formula of the probability of an event is written as,
$P\left( A \right) = \dfrac{{{\text{Number of favourable outcome}}}}{{{\text{Total Number of favourable outcome}}}}$
Here, the number of favorable outcomes are $4$ and the total number of favorable outcomes are $20$.
So, the probability that the number on the card is a perfect square is calculated as,
$P\left( A \right) = \dfrac{{\text{4}}}{{{\text{20}}}} \\\ = \dfrac{1}{5} \\\$
Therefore, the probability that the number on the card is a perfect square is $\dfrac{1}{5}$.

Note: Probability of an event will always be in between 0 to 1. If the probability of an event to occur is P, then the probability of the same event note to occur is 1-P.
Let P(E) be the probability of an event occurring and P’(E) be the probability of the event to not occur then $P(E)+P’(E)=1.$