To find the probability of occurrence of neither event A nor event B, we first need to understand that A and B are mutually exclusive events. This means that both events cannot occur at the same time. Therefore, the probability of occurrence of either A or B is given by the sum of their individual probabilities.
The probability of occurrence of event A, \(P(A) = \frac{1}{2}\), and the probability of occurrence of event B, \(P(B) = \frac{3}{10}\).
Since A and B are mutually exclusive, the probability of occurrence of either A or B is:
\(P(A \cup B) = P(A) + P(B)\)
Substituting the values, we get:
\(P(A \cup B) = \frac{1}{2} + \frac{3}{10}\)
To add these fractions, we need a common denominator:
\(\frac{1}{2} = \frac{5}{10}\)
Hence,
\(P(A \cup B) = \frac{5}{10} + \frac{3}{10} = \frac{8}{10} = \frac{4}{5}\)
The probability of occurrence of neither A nor B is the complement of the probability of occurrence of either A or B:
\(P(\text{neither } A \text{ nor } B) = 1 - P(A \cup B)\)
Substituting the calculated result:
\(P(\text{neither } A \text{ nor } B) = 1 - \frac{4}{5} = \frac{1}{5}\)
Therefore, the probability of occurrence of neither A nor B is \(\frac{1}{5}\). Hence, the correct answer is:
\(\frac{1}{5}\)
The correct option is: Option 4: \(\frac{1}{5}\)