Question

the probability for event a is 0.4, the probability for event b is 0.2, and the probability of events a and b is 0.1. why are the events not independent? the sum of p(a) and p(b) is greater than p(a and b). the product of p(a) and p(b) is greater than p(a and b). the product of p(a) and p(b) is not equal to p(a and b). the sum of p(a) and p(b) is not equal to p(a and b).

Explanation:

Step1: Definir la condición para eventos independientes

Para eventos independientes \(A\) y \(B\), \(P(A\cap B)=P(A)\times P(B)\).

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

Dado \(P(A) = 0.4\) y \(P(B)=0.2\), entonces \(P(A)\times P(B)=0.4\times0.2 = 0.08\).

Step3: Comparar con \(P(A\cap B)\)

Dado \(P(A\cap B) = 0.1\), y \(0.08
eq0.1\), es decir \(P(A)\times P(B)
eq P(A\cap B)\), por lo que los eventos no son independientes.

Answer:

C. The product of \(P(A)\) and \(P(B)\) is not equal to \(P(A \text{ and } B)\).