Let's solve the problem step by step.
### Step 1: Identify the Problem
We need to find the probability that a randomly selected number between 1 and 20 is the square of a natural number.
### Step 2: Identify Which Numbers Are Squares of Natural Numbers
First, we list the natural numbers whose squares could potentially fall between 1 and 20. These are the numbers 1, 2, 3, and 4 because:
- [tex]\(1^2 = 1\)[/tex]
- [tex]\(2^2 = 4\)[/tex]
- [tex]\(3^2 = 9\)[/tex]
- [tex]\(4^2 = 16\)[/tex]
These are all the squares of natural numbers less than or equal to [tex]\( \sqrt{20} \)[/tex], which is approximately 4.47. So, the relevant squares are 1, 4, 9, and 16.
### Step 3: Count the Number of Favorable Outcomes
The favorable outcomes are the numbers 1, 4, 9, and 16. So, there are 4 favorable outcomes.
### Step 4: Count the Total Number of Possible Outcomes
The numbers between 1 and 20 are inclusive, so there are 20 possible outcomes in total.
### Step 5: Calculate the Probability
The probability [tex]\( P \)[/tex] is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
[tex]\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{4}{20}
\][/tex]
### Step 6: Simplify the Probability
We simplify [tex]\(\frac{4}{20}\)[/tex] to get:
[tex]\[
\frac{4}{20} = \frac{1}{5} = 0.2
\][/tex]
### Conclusion
The probability that the number selected is the square of a natural number is [tex]\( \frac{1}{5} \)[/tex] or 0.2.
So, the correct answer is:
D. [tex]\(\frac{1}{5}\)[/tex]
