3.2 Exercise 8

Let event A = learning Spanish. Let event B = learning German. Then $A \cap B =$ learning Spanish and German. Suppose $P \left(A\right) = 0.4$ and $P \left(B\right) = 0.2$. $P \left(A \cap B\right) = 0.08$. Are events A and B independent? Hint: You must show ONE of the following:

• $P \left(\right. A \left|\right. B \left.\right) = P \left(A\right)$
• $P \left(\right. B \left|\right. A \left.\right) = P \left(B\right)$
• $P \left(A \cap B\right) = P \left(A\right) P \left(B\right)$