Question

select the correct answer. the venn diagram shows event a and event b comprised of outcomes from the same sample space. the probability of event a is given, as well as the probability of neither event a nor event b. what is the probability of event b? a. 0.5 b. 0.4 c. 0.2 d. 0.6

Explanation:

Step1: Recall probability formula

Let \(P(A) = 0.2\), and let the probability of neither \(A\) nor \(B\) be \(P(\overline{A\cup B})=0.6\). We know that \(P(A\cup B)+P(\overline{A\cup B}) = 1\). So, \(P(A\cup B)=1 - P(\overline{A\cup B})\).

Step2: Calculate \(P(A\cup B)\)

Since \(P(\overline{A\cup B}) = 0.6\), then \(P(A\cup B)=1 - 0.6=0.4\).

Step3: Use addition - rule for mutually - exclusive events

Since \(A\) and \(B\) are mutually - exclusive (disjoint, as there is no overlap in the Venn diagram), \(P(A\cup B)=P(A)+P(B)\). We know \(P(A) = 0.2\) and \(P(A\cup B)=0.4\). Then \(P(B)=P(A\cup B)-P(A)\).

Step4: Calculate \(P(B)\)

Substitute \(P(A\cup B) = 0.4\) and \(P(A)=0.2\) into the formula \(P(B)=P(A\cup B)-P(A)\), we get \(P(B)=0.4 - 0.2=0.2\).

Answer:

C. 0.2