Probability trees and Venn diagrams

Probability trees

• A tree diagram maps out events that happen in stages (two coin tosses, two counters drawn).
• Each branch is labelled with the probability of one outcome; the branches from any one point add to 1.
• The two rules:
  • Multiply along a path for a combined outcome — an "and" (this then that).
  • Add separate paths that each count — an "or" (this path or that path).

With and without replacement

• With replacement: the item is put back, so the totals are unchanged and the probabilities stay the same at each stage. The stages are independent.
• Without replacement: the item is kept, so the total shrinks and later probabilities change — both the count you want and the total drop by one.

Venn diagrams

• A Venn diagram shows events as overlapping circles inside a rectangle (all outcomes).
• The overlap (intersection) is "$A$ and $B$" — outcomes in both.
• Everything in either circle is "$A$ or $B$" (the union).
• Mutually exclusive events cannot both happen, so their circles do not overlap ($P(A \text{ and } B) = 0$).
• Independent events do not affect each other: $P(A \text{ and } B) = P(A) \times P(B)$. (Independent is not the same as mutually exclusive.)