Question

1. for two independent events, f and g, it is known that ( p(f)=0.8 ) and ( p(f \text { and } g)=0.24 ). which of the following statements is true? a) ( p(g)=0.06 ) b) ( p(g)=0.20 ) c) ( p(f | g)=0.30 ) d) ( p(f \text { or } g)=0.86 ) e) ( p(f \text { or } g)=1 )

Explanation:

Step1: Use the formula for independent events

For independent events \(F\) and \(G\), \(P(F\cap G)=P(F)\times P(G)\). Given \(P(F) = 0.8\) and \(P(F\cap G)=0.24\), we can find \(P(G)\) as \(P(G)=\frac{P(F\cap G)}{P(F)}=\frac{0.24}{0.8}=0.3\).

Step2: Use the formula for conditional probability

The formula for conditional probability is \(P(F|G)=\frac{P(F\cap G)}{P(G)}\). Since \(P(F\cap G) = 0.24\) and \(P(G)=0.3\) (from Step 1), then \(P(F|G)=\frac{0.24}{0.3}=0.8\) (this is wrong for option C as we'll see).

Step3: Use the formula for \(P(F\cup G)\)

The formula for \(P(F\cup G)\) is \(P(F\cup G)=P(F)+P(G)-P(F\cap G)\). Substitute \(P(F) = 0.8\), \(P(G)=0.3\) and \(P(F\cap G)=0.24\) into the formula: \(P(F\cup G)=0.8 + 0.3-0.24=0.86\)

Answer:

D. \(P(F\ or\ G)=0.86\)