Question

A card is drawn from a packet of 50 identical cards numbered from 1 to 50. The probability of drawing a number which is a perfect square, is :

• A. \(\frac{7}{50}\)
• B. \(\frac{8}{50}\)
• C. \(\frac{6}{50}\)
• D. \(\frac{10}{50}\)

Hint:

To find the number of perfect squares up to any number \(N\), simply calculate \(\lfloor\sqrt{N}\rfloor\) (the floor value of the square root of \(N\)).
For \(N = 50\):
\[ \sqrt{50} \approx 7.07 \implies \lfloor 7.07 \rfloor = 7 \]
This gives you the count of perfect squares as 7 instantly, without having to write out the sequence!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
There are 50 cards numbered from 1 to 50. One card is drawn at random. We need to find the probability that the card drawn has a number that is a perfect square.

Step 2: Key Formula or Approach:
The classical probability \(P(E)\) of an event \(E\) is defined as:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

Step 3: Detailed Explanation:
1. Find the total number of possible outcomes:
Since the cards are numbered from 1 to 50, the total number of cards is:
\[ \text{Total outcomes} = 50 \]
2. Identify the perfect square numbers between 1 and 50:
A perfect square is an integer that can be written as the square of another integer. Let's list them:
\[ 1^2 = 1 \]
\[ 2^2 = 4 \]
\[ 3^2 = 9 \]
\[ 4^2 = 16 \]
\[ 5^2 = 25 \]
\[ 6^2 = 36 \]
\[ 7^2 = 49 \]
The next perfect square is \(8^2 = 64\), which is greater than 50.
3. Write down the set of favorable numbers:
\[ \text{Favorable outcomes} = \{1, 4, 9, 16, 25, 36, 49\} \]
Count of favorable outcomes is 7.
4. Calculate the probability of drawing a perfect square:
\[ P(\text{Perfect Square}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{7}{50} \]

Step 4: Final Answer:
The probability of drawing a perfect square is \(\frac{7}{50}\), which corresponds to option (A).