Step-by-step Solution:

We are given \( n \) events \( E_1, E_2, \dots, E_n \) with respective probabilities \( p_1, p_2, \dots, p_n \). The given condition states: \[ P(E_1, E_2, \dots, E_n) = \prod_{i=1}^{n} p_i \] Step 1: Understanding the given condition The probability of the simultaneous occurrence of multiple events is given by the product of their individual probabilities. This is the definition of independent events. Mathematically, events \( E_1, E_2, \dots, E_n \) are independent if: \[ P(E_1 \cap E_2 \cap \dots \cap E_n) = P(E_1) P(E_2) \dots P(E_n) \] This matches exactly with the given condition. Step 2: Evaluating the options - \[Option A:\] "The events are mutually exclusive" - Mutually exclusive events satisfy \( P(E_i \cap E_j) = 0 \) for \( i \neq j \), meaning they cannot occur together. - Since our given equation states that the probability of their simultaneous occurrence is nonzero, they are not mutually exclusive. - Thus, this option is incorrect. - \[Option B:\] "The events are independent" - As derived above, the given probability condition confirms independence - Thus, this option is correct. - \[Option C:\] "The events are dependent" - Since we proved the events are independent, this option is incorrect. - \[Option D:\] "The events are mutually exclusive and independent" - If events were mutually exclusive, the probability of their intersection would be zero. - Since they are independent, they cannot be mutually exclusive unless one of them has zero probability. - Thus, this option is incorrect. Final Answer: \[ \boxed{\text{The events are independent.}} \]