To determine whether the events "buy popcorn" (let's call it \(A\)) and "buy juice" (let's call it \(B\)) are independent, we need to verify if the following condition holds:
\[ P(A \cap B) = P(A) \times P(B) \]
Where:
- \(P(A)\) is the probability of buying popcorn.
- \(P(B)\) is the probability of buying juice.
- \(P(A \cap B)\) is the probability of buying both popcorn and juice.
We are given:
- \(P(A) = 0.5\) (the probability Filip buys popcorn)
- \(P(B) = 0.4\) (the probability Filip buys juice)
- \(P(A \cap B) = 0.2\) (the probability Filip buys both popcorn and juice)
Now let's compute \(P(A) \times P(B)\):
\[ P(A) \times P(B) = 0.5 \times 0.4 = 0.2 \]
Now we compare \(P(A \cap B)\) with \(P(A) \times P(B)\):
- \(P(A \cap B) = 0.2\)
- \(P(A) \times P(B) = 0.2\)
Since \(P(A \cap B) = P(A) \times P(B)\), the two events are independent.
Therefore, the answer is 2 for independent.