Question

Cards marked with numbers 13, 14, 15, ……., 60 are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that number on the card drawn is
(i) divisible by 5
(ii) a number is a perfect square

Answer 1

Hint – We will start solving this question by writing down all the numbers with which the cards are marked. Then by using the formula of probability of an event, i.e., Probability = Number of favorable outcomes / Total number of outcomes, we will find the solution for both the parts.

Complete step-by-step answer:
Given that the cards are marked with the numbers from 13 - 60, i.e.,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22 ,23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, and 60.
As, the numbers have been started form 13, therefore,
Total number of outcomes = 60 – 12
                                                 = 48
(i) divisible by 5.
Let $E_1$ be the event of getting a number divisible by 5.
The given numbers are,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22 ,23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, and 60.
Numbers which are divisible by 5 are,
15, 20, 25, 30, 35, 40, 45, 50, 55, and 60.
Thus, number of outcomes favorable to $E_1$= 10
$\therefore$Probability ($E_1$) = ${\displaystyle \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}}$
$$\begin{gathered} \Rightarrow P\left(E_1\right)={\displaystyle \frac{10}{48}} \\ \Rightarrow P\left(E_1\right)={\displaystyle \frac{5}{24}} \end{gathered}$$
Hence, the probability of getting a number divisible by 5 is ${\displaystyle \frac{5}{24}}$.
(ii) a number is a perfect square
Let $E_2$ be the event of getting a number which is a perfect square.
The given numbers are,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22 ,23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, and 60.
Numbers which are a perfect square are,
16, 25, 36 and 49.
Thus, number of outcomes favorable to $E_2$= 4
$\therefore$Probability $\left(E_2\right)$ = ${\displaystyle \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}}$
$$\begin{gathered} \Rightarrow P\left(E_2\right)={\displaystyle \frac{4}{48}} \\ \Rightarrow P\left(E_2\right)={\displaystyle \frac{1}{12}} \end{gathered}$$
Hence, the probability of getting a number which is a perfect square, is ${\displaystyle \frac{1}{12}}$.

Note – The probability of an event is the likelihood of that event occurring. It is a value between and including zero and one. These kinds of questions are very simple if one knows the basic formulas and the formulas must be remembered as there is no alternate method for solving this question.