Adding alternatives and distinguishing independence

Mutually exclusive events cannot occur together. For these events, \(\mathrm P(A\cup B)=\mathrm P(A)+\mathrm P(B)\). If events overlap, subtract the overlap once: \(\mathrm P(A\cup B)=\mathrm P(A)+\mathrm P(B)-\mathrm P(A\cap B)\).

Example

On one fair die, \(A\) is “even” and \(B\) is “greater than four”. The sets are \(\{2,4,6\}\) and \(\{5,6\}\), sharing \(6\). Thus \(\mathrm P(A\cup B)=\dfrac36+\dfrac26-\dfrac16=\dfrac46=\dfrac23\).

Independent events do not change each other’s probabilities. Then \(\mathrm P(A\cap B)=\mathrm P(A)\mathrm P(B)\). For independent fair dice, “first die even” and “second die six” have joint probability \(\dfrac12\times\dfrac16=\dfrac1{12}\).

Independence and mutual exclusivity are different. “First die is one” and “first die is six” cannot occur together; knowing the first happens makes the second impossible, so these positive-probability events are not independent.