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

Question

"buy juice" are independent events. Enter 1 for dependent or 2 for independent. (1 point)

Answer 1

To determine whether the events are independent, we can use the definition of independent events. Two events A and B are independent if:

\[ P(A \cap B) = P(A) \times P(B) \]

In this scenario, let:

• Event A = "buy popcorn"
• Event B = "buy juice"

From the information given:

• \( P(A) = 0.5 \) (the probability of buying popcorn)
• \( P(B) = 0.4 \) (the probability of buying juice)
• \( P(A \cap B) = 0.2 \) (the probability of buying both popcorn and juice)

Now we can calculate \( P(A) \times P(B) \):

\[ P(A) \times P(B) = 0.5 \times 0.4 = 0.2 \]

Now we compare \( P(A \cap B) \) and \( P(A) \times P(B) \):

\[ P(A \cap B) = 0.2 \] \[ P(A) \times P(B) = 0.2 \]

Since \( P(A \cap B) = P(A) \times P(B) \), we conclude that the events "buy popcorn" and "buy juice" are independent.

So, the final answer is 2 for independent events.