To determine the probability of the complement of an event, we must understand the relationship between an event and its complement.
1. Understanding Complementary Probabilities:
- The probability of an event [tex]\(A\)[/tex], denoted as [tex]\(P(A)\)[/tex], and the probability of its complement, [tex]\(P(A^c)\)[/tex], always sum to 1. That is:
[tex]\[
P(A) + P(A^c) = 1
\][/tex]
- The complement of an event [tex]\(A\)[/tex], denoted [tex]\(A^c\)[/tex], represents all outcomes that are not in event [tex]\(A\)[/tex].
2. Given Probability of the Event:
- In this problem, the probability of the event [tex]\(A\)[/tex] is given as [tex]\( \frac{2}{7} \)[/tex].
3. Finding the Probability of the Complement:
- Using the relationship [tex]\(P(A) + P(A^c) = 1\)[/tex], we can solve for [tex]\(P(A^c)\)[/tex].
- Substituting the given probability:
[tex]\[
\frac{2}{7} + P(A^c) = 1
\][/tex]
- To find [tex]\(P(A^c)\)[/tex], subtract [tex]\(\frac{2}{7}\)[/tex] from both sides:
[tex]\[
P(A^c) = 1 - \frac{2}{7}
\][/tex]
4. Calculating the Result:
- We can express 1 as a fraction with a denominator of 7 for easier subtraction:
[tex]\[
1 = \frac{7}{7}
\][/tex]
- Subtracting the fractions:
[tex]\[
P(A^c) = \frac{7}{7} - \frac{2}{7} = \frac{5}{7}
\][/tex]
5. Conclusion:
- So, the probability of the complement of the event is:
[tex]\[
P(A^c) = \frac{5}{7}
\][/tex]
Answer:
[tex]\(\frac{5}{7}\)[/tex]
