Mathematics

Mathematics

Problem

Two coins are tossed simultaneously. Which one of the following collections of events forms a partition of the sample space? a) {at least one Head,  exactly one Tail}\{\text{at least one Head},\;\text{exactly one Tail}\} b) {exactly two Heads,  at most one Tail}\{\text{exactly two Heads},\;\text{at most one Tail}\} c) {first coin shows Head,  first coin shows Tail}\{\text{first coin shows Head},\;\text{first coin shows Tail}\} d) {both coins show the same face,  first coin shows Head}\{\text{both coins show the same face},\;\text{first coin shows Head}\}
Intuitive explanation

A partition means the events must be mutually exclusive (no overlap) and collectively exhaustive (cover everything together). The key insight is that you're looking for a set of events where every possible outcome lands in exactly one event—no gaps, no double-counting. Think about what outcomes are possible when tossing two coins, then check which option divides them cleanly into non-overlapping groups that account for all possibilities.

Intuitive explanation

A partition means the events must be mutually exclusive (no overlap) and collectively exhaustive (cover everything together). The key insight is that you're looking for a set of events where every possible outcome lands in exactly one event—no gaps, no double-counting. Think about what outcomes are possible when tossing two coins, then check which option divides them cleanly into non-overlapping groups that account for all possibilities.

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