SL 4.6—Combined, conditional and independent events

Mutually exclusive events cannot occur together, so P(A∩B)=0. Independent events do not change one another's probability, so P(A∩B)=P(A)P(B) and P(A|B)=P(A) when defined.

Two non-impossible mutually exclusive events are automatically dependent: learning that A occurred makes B impossible. Independent events can occur together, such as separate results from a fair coin and die.

For a fair coin and die, P(head and six)=1/2×1/6=1/12. For one die roll, ‘odd’ and ‘even’ are mutually exclusive and their intersection is zero; they are not independent because P(odd|even)=0.

‘Independent’ does not mean unrelated in every philosophical sense, and ‘exclusive’ does not mean equally likely. Use the intersection or conditional rule that matches the claim.

Use a Venn diagram for overlapping sets, a tree for sequential conditions, and a table or sample-space diagram for paired outcomes. Without replacement, branch probabilities change because the remaining total and composition change; with replacement they reset. Independence must be checked from $P(A\cap B)=P(A)P(B)$ or $P(A|B)=P(A)$, not assumed from the wording.