To find the probability of selecting a student with blue eyes, we first need to determine the total number of students surveyed. We can calculate this by summing the number of students for each eye color:
- Blue: 20
- Brown: 12
- Green: 10
- Amber: 8
- Hazel: 6
- Gray: 4
Now, we sum these values:
\[ 20 + 12 + 10 + 8 + 6 + 4 = 70 \]
Thus, the total number of students is 70.
Next, we can calculate the probability of selecting a student with blue eyes. The number of students with blue eyes is 20. The probability \(P\) of selecting a student with blue eyes can be calculated using the formula:
\[ P(\text{blue eyes}) = \frac{\text{Number of students with blue eyes}}{\text{Total number of students}} = \frac{20}{70} \]
To simplify the fraction \(\frac{20}{70}\), we divide both the numerator and the denominator by their greatest common divisor, which is 10:
\[ \frac{20 \div 10}{70 \div 10} = \frac{2}{7} \]
Therefore, the probability of randomly selecting a student with blue eyes is:
\[ \frac{2}{7} \]
So the answer in simplest form is \(\frac{2}{7}\).