Question #41012

This question targets the conceptual collision between independent trials and mutually exclusive sub-events within a binomial framework.

The trap lies in assuming that the independence of individual coin tosses automatically simplifies the intersection of macroscopic events, or incorrectly applying the addition rule $P(A \cup B)$ instead of rigorously evaluating the intersection $P(A \cap S)$.

Students often fail to correctly partition the sample space for $A \cap S$, forgetting that fixing the first trial restricts the binomial general term to the remaining $n-1$ trials.

By forcing the algebraic expansion of the independence condition $P(A \cap S) = P(A)P(S)$, the problem tests whether the student can accurately navigate localized sample space reductions without violating the fundamental definitions of probability.