If P(E) = 0.6 and P(E|F) = 0.34, are events E and F independent?

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. Use the conditional probability formula to express \(P(E \cap F)\) in terms of \(P(E|F)\) and \(P(F)\): \(P(E \cap F) = P(E|F) \times P(F)\).
3. Substitute the given value \(P(E|F) = 0.34\) into the equation: \(P(E \cap F) = 0.34 \times P(F)\).
4. Since \(P(E)\) is given as 0.6, write the independence condition as \$0.34 \times P(F) = 0.6 \times P(F)$.
5. Analyze the equation to determine if it can hold true for any \(P(F)\) between 0 and 1. If it cannot, conclude that E and F are not independent.

Key Concepts

Probability of an Event

The probability of an event, denoted P(E), measures the likelihood that the event E will occur. It ranges from 0 to 1, where 0 means impossible and 1 means certain. Understanding this helps in comparing probabilities and analyzing relationships between events.

Conditional Probability

Conditional probability, P(E|F), is the probability of event E occurring given that event F has already occurred. It reflects how the occurrence of one event affects the likelihood of another, which is crucial for assessing dependence between events.

Independence of Events

Two events E and F are independent if the occurrence of one does not affect the probability of the other. Mathematically, this means P(E|F) = P(E). Checking this equality helps determine if events influence each other or not.