To determine the probability of both independent events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] occurring, denoted as [tex]\(P(A \text{ and } B)\)[/tex], we need to use the rule for the probability of independent events. The probability that both event [tex]\(A\)[/tex] and event [tex]\(B\)[/tex] occur is the product of their individual probabilities.
Given:
- [tex]\(P(A) = 0.50\)[/tex]
- [tex]\(P(B) = 0.20\)[/tex]
Since [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent events, the formula to calculate [tex]\(P(A \text{ and } B)\)[/tex] is:
[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]
Substituting the given probabilities into the formula:
[tex]\[ P(A \text{ and } B) = 0.50 \times 0.20 \][/tex]
Performing the multiplication:
[tex]\[ P(A \text{ and } B) = 0.10 \][/tex]
So, the probability that both events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] occur is:
[tex]\[ P(A \text{ and } B) = 0.10 \][/tex]
Thus, the correct answer is:
A. 0.10
