Suppose that P(E) = 0.8, P(F) = 0.5, and P(E and F) = 0.24. Are events E and F independent? Why?

Verified step by step guidance

1. Recall the definition of independence for two events E and F: E and F are independent if and only if \(P(E \cap F) = P(E) \times P(F)\).
2. Identify the given probabilities: \(P(E) = 0.8\), \(P(F) = 0.5\), and \(P(E \cap F) = 0.24\).
3. Calculate the product of \(P(E)\) and \(P(F)\): compute \(P(E) \times P(F) = 0.8 \times 0.5\).
4. Compare the calculated product \(P(E) \times P(F)\) with the given \(P(E \cap F)\) to check if they are equal.
5. If \(P(E \cap F) = P(E) \times P(F)\), then E and F are independent; otherwise, they are dependent.

Key Concepts

Probability of Events

Probability measures the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain). For events E and F, P(E) and P(F) represent their individual probabilities, while P(E and F) represents the probability that both events occur simultaneously.

Independence of Events

Two events E and F are independent if the occurrence of one does not affect the probability of the other. Mathematically, E and F are independent if and only if P(E and F) = P(E) × P(F). This condition helps determine if events influence each other.

Checking Independence Using Given Probabilities

To check if E and F are independent, calculate P(E) × P(F) and compare it to P(E and F). If they are equal, the events are independent; if not, they are dependent. In this question, comparing 0.8 × 0.5 = 0.4 with 0.24 shows dependence.