Question

Let sigma = {a,b}. choose any two languages

Answer 1

The events A and B are independent because the probability of their conjunction is equal to the product of their individual probabilities.

Step-by-step explanation:

Event A represents the probability of learning Spanish, while event B represents the probability of learning German. The events A and B are independent if the probability of their conjunction (A AND B) is equal to the product of their individual probabilities, P(A) and P(B), i.e., P(A AND B) = P(A) x P(B).

In this case, P(A) = 0.4 and P(B) = 0.2, and P(A AND B) = 0.08. Let's check if the events are independent:

1. Multiply the individual probabilities: 0.4 x 0.2 = 0.08
2. Compare the result with P(A AND B): 0.08 == 0.08

Since the probabilities are equal, events A and B are independent.