Question

let a and b be two events with the following probabilities.

• ( p(a)=0.4 )
• ( p(b)=0.2 )
• ( p(a \text{ and } b)=0.16 )

complete the sentence.
events a and b must independent because

Explanation:

Step1: Recall the formula for independent events

For two independent events \(A\) and \(B\), \(P(A\cap B)=P(A)\times P(B)\).

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

Given \(P(A) = 0.4\) and \(P(B)=0.2\), then \(P(A)\times P(B)=0.4\times0.2 = 0.08\).

Step3: Compare \(P(A\cap B)\) with \(P(A)\times P(B)\)

We have \(P(A\cap B) = 0.16\) and \(P(A)\times P(B)=0.08\). Since \(0.16
eq0.08\), events \(A\) and \(B\) are not independent.

Answer:

Events \(A\) and \(B\) must not be independent because \(P(A\cap B)
eq P(A)\times P(B)\) (specifically \(0.16
eq0.4\times0.2\)).