Probability

Some cards are numbered from $10$ to $40$. They are well shuffled and one card is drawn at random. What is the probability that the card drawn is:

1. a prime number;


2. divisible by $2$ and $5$;


3. a perfect square.

Sample space $= S = \left\{10, 11, 12, \cdots, 40 \right\}$

$\therefore \;$ Number of elements in sample space $= n \left(S\right) = 31$

1. Let $A =$ event that the number on the card is a prime number
   
   Then $A = \left\{11, 13, 17, 19, 23, 29, 31, 37 \right\}$
   
   $\therefore \;$ Number of elements in $A = n \left(A\right) = 8$
   
   $\therefore \;$ Probability of event $A = P \left(A\right) = \dfrac{n \left(A\right)}{n \left(S\right)} = \dfrac{8}{31}$


2. Let $B =$ event that the number on the card drawn is divisible by $2$ and $5$
   
   Then $B = \left\{10, 20, 30, 40 \right\}$
   
   $\therefore \;$ Number of elements in $B = n \left(B\right) = 4$
   
   $\therefore \;$ Probability of event $B = P \left(B\right) = \dfrac{n \left(B\right)}{n \left(S\right)} = \dfrac{4}{31}$


3. Let $C =$ event that the number on the card is a perfect square
   
   Then $C = \left\{16, 25, 36 \right\}$
   
   $\therefore \;$ Number of elements in $C = n \left(C\right) = 3$
   
   $\therefore \;$ Probability of event $C = P \left(C\right) = \dfrac{n \left(C\right)}{n \left(S\right)} = \dfrac{3}{31}$