To determine whether events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent, we need to verify if the following condition holds:
[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]
We are given:
- [tex]\( P(A) = \frac{1}{2} \)[/tex]
- [tex]\( P(B) = \frac{1}{5} \)[/tex]
- [tex]\( P(A \text{ and } B) = \frac{1}{10} \)[/tex]
First, let's calculate [tex]\( P(A) \times P(B) \)[/tex]:
[tex]\[ P(A) \times P(B) = \left( \frac{1}{2} \right) \times \left( \frac{1}{5} \right) = \frac{1}{2} \times \frac{1}{5} = \frac{1}{10} \][/tex]
Now, compare this result with [tex]\( P(A \text{ and } B) \)[/tex]:
[tex]\[ P(A \text{ and } B) = \frac{1}{10} \][/tex]
Since [tex]\( P(A) \times P(B) = \frac{1}{10} \)[/tex] and [tex]\( P(A \text{ and } B) = \frac{1}{10} \)[/tex], we see that:
[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]
Therefore, events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent.
Hence, the statement "A and B are independent events." is True.
