Question

A bag contains 25 cards numbered from 1 to 25. A card is drawn at random from the bag. Find the probability that the number on the drawn card is:
(i) Divisible by 3 or 5
(ii) Perfect square number

Answer 1

Hint: As per the divisibility rules, find the number which are divisible by 3 and the numbers which are divisible by 5. Then, find the probability required. Find the perfect number of squares between 1 to 25 and thus, find the probability.

Complete step by step answer:
Given is a bag containing 25 cards numbered from 1 to 25. From which a card is drawn randomly. We need to find the probability that the number on the drawn card is divisible by 3 or 5 and also it is a perfect square number.

We know that, the probability of drawing a card at random from the bag containing 25 cards would be: \[\dfrac{1}{25}\]
First, we have to find out that the drawn card is divisible by 3 or 5. It can be divisible by either 3 or 5.
Divisibility rule for 3: a number is divisible by 3 only if the sum of all the digits is divisible by 3.
For example: \[\text{21 }=\text{ 2}+\text{1 }=\text{ 3}\] 3 is divisible by 3, so 21 is divisible by 3.
Divisibility rule for 5: end digits of any number must be 0 or 5.
For example: \[\text{1}0,\text{ 15},\text{ 25 etc}.\]
Now, the number from 1 to 25 which are divisible by 3 are:
\[\text{3},\text{ 6},\text{ 9},\text{ 12},\text{ 15},\text{ 18},\text{ 21},\text{ 24 }=\text{ 8 numbers total}\]
Now, the number from 1 to 25 which are divisible by 5 are:
\[\text{5},\text{ 1}0,\text{ 15},\text{ 2}0,\text{ 25 }=\text{ 5 numbers in total}\]
Suppose, the probability of drawn card to be divisible by 3 is P (A).
Then, \[P(A)=\dfrac{8}{25}\]
If the probability of drawn card to be divisible by 5 is P (B).
Then, \[P(B)=\dfrac{5}{25}\]
But we need to find the probability of drawn card to be divisible by either 5 or 3 i.e. P (A or B) we have formula:
\[P(A\text{ or B)=P(A)+P(B)-P(A and B)}\]
Where, P (A and B) = probability of number divisible by both 3 and 5, which is $\dfrac{1}{25}$ here, as 15 can be divided by both 3 and 5.
\[\begin{align}
  & P(A\text{ or B)=}\dfrac{8}{25}+\dfrac{5}{25}-\dfrac{1}{25} \\
 & \Rightarrow P(A\text{ or B)=}\dfrac{12}{25} \\
\end{align}\]
Now, we have to find the probability of drawn card to be a perfect square:
Suppose it in P (C).
The number from 1-25 which are perfect square:
\[\text{1},\text{ 4},\text{ 9},\text{ 16},\text{ 25 }=\text{ 5}\]
Therefore,
\[\begin{align}
  & P(C)=\dfrac{5}{25}=\dfrac{1}{5} \\
 & \Rightarrow P(C)=\dfrac{1}{5} \\
\end{align}\]

Note: We need to consider all the numbers from 1 to 25. If the question mentioned to consider the number between 1 to 25, then the answer would have been different, because in that case we would not consider 1 and 25 to be included.