Question

can two events with nonzero probabilities be both independent and mutually exclusive?

choose the correct answer below.

a. no, two events with nonzero probabilities cannot be independent and mutually exclusive because independence is the complement of being mutually exclusive.
b. yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities are equal.
c. no, two events with nonzero probabilities cannot be independent and mutually exclusive because if two events are mutually exclusive, then when one of them occurs, the probability of the other must be zero.
d. yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities add up to one.

Explanation:

Define mutually exclusive events

Using the Mutually Exclusive Events knowledge point

$$ P(A \cap B) = 0 $$

Define independent events

Using the Independent Events Probability knowledge point

$$ P(A \cap B) = P(A) \cdot P(B) $$

Analyze the contradiction for nonzero probabilities

Using the Independent Events Probability and Mutually Exclusive Events knowledge points

$$
LATEXBLOCK0
$$

Connect to the conceptual explanation

If two events are mutually exclusive, the occurrence of one event means the other cannot occur. Therefore, if event \(A\) occurs, the probability of event \(B\) occurring becomes zero, which changes its probability from its original nonzero value \(P(B)\). This dependency means they cannot be independent.

Answer:

• A. No, two events with nonzero probabilities cannot be independent and mutually exclusive because independence is the complement of being mutually exclusive.
• B. Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities are equal.
• C. No, two events with nonzero probabilities cannot be independent and mutually exclusive because if two events are mutually exclusive, then when one of them occurs, the probability of the other must be zero. (Correct answer)
• D. Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities add up to one.