Question

Cards, marked with numbers 5 to 50, are placed in a box and mixed throughly. A card is drawn from teh box at random. Find the probability that the number on the taken out card is: (i) a prime number less than 10. (ii) a number which is a perfect square.

Answer 1

The numbers are 5,6,7,8,9,10............
prime numbers less than 10=5,7
perfect squares = 9,16,25,36,49

Answer 2

$\underline{\underline{\Huge{\textbf{Solution:}}}}$

Given,
$\textit{Experiment -}$ A card is drawn from the box containing numbers from 5 to 50 at random.

Let $E_{1}$ be the Event that drawn card is a prime number less than 10

and $E_{2}$ be the Event that drawn card is a number which which is a Perfect square.

P($E_{1}$) = ?

P($E_{2}$) = ?



$\underline{\large{\textsf{Step-1:}}}$
Find the Total number of outcomes when a card is drawn from the box.

Total No. of Outcomes = No. of ways of drawing a card from the total cards in the box.

Total No. of cards in the box = 46 (50-5+1)

Total No. of Outcomes = No. of ways of drawing a card from 46 cards

$\therefore$ Total No. of outcomes = 46.



$\underline{\large{\textsf{Step-2:}}}$
Find the Number of Favorable outcomes for Event $E_{1}[n(E_{1})]$

No. of Favorable outcomes for Event $E_{1}$ = No. of Prime Numbers from 5 to 10

$E_{1}$ = {5, 7}
No. of Prime Numbers from 5 to 10 = 2

$\therefore$ No. of Favorable outcomes for Event $E_{1}, n(E_{1})$ = 2



$\underline{\large{\textsf{Step-3:}}}$
Find the Number of Favorable outcomes for Event $E_{2}[n(E_{2})]$

No. of Favorable outcomes for Event $E_{1}$ = No. of Perfect square from 5 to 50

$E_{2}$ = {9, 16, 25, 36, 49}
No. of Prime Numbers from 5 to 50 = 5

$\therefore$ No. of Favorable outcomes for Event $E_{2}, n(E_{2})$ = 5



$\underline{\large{\textsf{Step-4:}}}$
Find the Probabilities of Event $E_{1}$ & $E_{2}$

We have,
$\bigstar\: \mathbf{Probability\: of\: occurrence\: of\: an\: Event= \dfrac{No.\: of\: Favorable\: Outcomes\: for\: Event}{Total\: No.\: of\: Outcomes\: of\: Experiment}}$

$P(E_{1}) = \dfrac{n(E_{1})}{n(S)}$

$\implies P(E_{1}) = \dfrac{2}{46}$

$\implies P(E_{1}) = \dfrac{1}{23}$

$P(E_{2}) = \dfrac{n(E_{2})}{n(S)}$

$\implies P(E_{2}) = \dfrac{5}{46}$



$\blacksquare \: \textsf{Probability that number on the card drawn is a Prime number less than 10 is \underline{\large{\mathbf{ \dfrac{1}{23} }}}}$
$\blacksquare \: \textsf{Probability that number on the card drawn is a Perfect square is \underline{\large{\mathbf{ \dfrac{5}{46} }}}}$