Question

Filip goes to a movie theater on weekends. On any given weekend, there is a 50% chance he will buy popcorn, there is a 40% chance he will buy juice, and there is a 20% chance he will buy both popcorn and juice. Mathematically determine whether the two events “buy popcorn” and “buy juice” are independent events. Enter 1 for dependent or 2 for independent.

Answer 1

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.