True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false,
explain why.
4. When two events are independent, they are also mutually exclusive.

Verified step by step guidance

1. Step 1: Begin by understanding the definitions of 'independent events' and 'mutually exclusive events'. Independent events are those where the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this is expressed as P(A ∩ B) = P(A) * P(B). Mutually exclusive events, on the other hand, are events that cannot occur at the same time. Mathematically, this means P(A ∩ B) = 0.
2. Step 2: Analyze the relationship between these two concepts. If two events are mutually exclusive, the probability of their intersection (both occurring simultaneously) is zero. However, for independent events, the probability of their intersection is generally non-zero unless one of the events has a probability of zero.
3. Step 3: Consider an example to clarify the distinction. Suppose Event A is 'rolling a 3 on a die' and Event B is 'rolling a 4 on the same die'. These events are mutually exclusive because you cannot roll both a 3 and a 4 in a single roll. However, they are not independent because the occurrence of one event does not affect the occurrence of the other—they are mutually exclusive instead.
4. Step 4: Reflect on whether independence and mutual exclusivity can coexist. If two events are mutually exclusive, the occurrence of one event means the other cannot occur, which violates the definition of independence. Therefore, mutually exclusive events cannot be independent.
5. Step 5: Conclude that the statement 'When two events are independent, they are also mutually exclusive' is false. Provide reasoning that independence and mutual exclusivity are distinct concepts and cannot occur simultaneously.

Key Concepts

Independent Events

Independent events are those whose occurrence or non-occurrence does not affect the probability of the other event occurring. For example, flipping a coin and rolling a die are independent events because the outcome of one does not influence the outcome of the other.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. For instance, when flipping a coin, the outcomes 'heads' and 'tails' are mutually exclusive because if one occurs, the other cannot. This concept is crucial in probability as it affects how we calculate the likelihood of combined events.

Relationship Between Independence and Mutual Exclusivity

The relationship between independent and mutually exclusive events is that they are fundamentally different. While independent events can occur simultaneously without affecting each other's probabilities, mutually exclusive events cannot occur together at all. Therefore, if two events are independent, they cannot be mutually exclusive.