To solve for the probability of the complement of an event, we need to understand a fundamental principle of probability: the sum of the probability of an event and the probability of its complement always equals 1. This is represented mathematically as:
[tex]\[ P(\text{event}) + P(\text{complement}) = 1 \][/tex]
Given that the probability of the event (let's denote it as [tex]\( P(\text{event}) \)[/tex]) is [tex]\(\frac{2}{7}\)[/tex], we want to find the probability of the complement of this event (denoted as [tex]\( P(\text{complement}) \)[/tex]).
Using the principle mentioned above, we can write:
[tex]\[ P(\text{event}) + P(\text{complement}) = 1 \][/tex]
Substituting [tex]\( P(\text{event}) = \frac{2}{7} \)[/tex] into the equation:
[tex]\[ \frac{2}{7} + P(\text{complement}) = 1 \][/tex]
To solve for [tex]\( P(\text{complement}) \)[/tex], we isolate it on one side of the equation by subtracting [tex]\(\frac{2}{7}\)[/tex] from both sides:
[tex]\[ P(\text{complement}) = 1 - \frac{2}{7} \][/tex]
Next, we need to perform the subtraction. To subtract [tex]\(\frac{2}{7}\)[/tex] from 1, we first express 1 as a fraction with a denominator of 7:
[tex]\[ 1 = \frac{7}{7} \][/tex]
Thus, the subtraction becomes:
[tex]\[ P(\text{complement}) = \frac{7}{7} - \frac{2}{7} \][/tex]
Since the denominators are the same, we simply subtract the numerators:
[tex]\[ P(\text{complement}) = \frac{7 - 2}{7} = \frac{5}{7} \][/tex]
So, the probability of the complement of the event is:
[tex]\[ \boxed{\frac{5}{7}} \][/tex]
Therefore, among the given choices, the correct answer is [tex]\(\frac{5}{7}\)[/tex].
