Question

event a has a 0.3 probability of occurring and event b has a 0.4 probability of occurring. a and b are independent events. what is the probability that neither a or b occurs? (1 point) 0.70 0.10 0.42 0.12

Explanation:

Step1: Calculate the probability of A not occurring

The probability of event A not occurring, denoted as \(P(\overline{A})\), is \(1 - P(A)\). Given \(P(A)=0.3\), then \(P(\overline{A})=1 - 0.3=0.7\).

Step2: Calculate the probability of B not occurring

The probability of event B not occurring, denoted as \(P(\overline{B})\), is \(1 - P(B)\). Given \(P(B) = 0.4\), then \(P(\overline{B})=1 - 0.4 = 0.6\).

Step3: Use the formula for independent events

Since A and B are independent, \(\overline{A}\) and \(\overline{B}\) are also independent. The probability that neither A nor B occurs is \(P(\overline{A}\cap\overline{B})\). For independent events, \(P(\overline{A}\cap\overline{B})=P(\overline{A})\times P(\overline{B})\). Substitute \(P(\overline{A}) = 0.7\) and \(P(\overline{B})=0.6\) into the formula: \(P(\overline{A}\cap\overline{B})=0.7\times0.6 = 0.42\).

Answer:

\(0.42\) (the third option)