Story Proofs, Axioms of Probability

Story Proof

• Proof by interpretation
  • Count the same thing in two ways:
    • $\binom{n}{k} = \binom{n}{n-k}$
    • $ k\binom{n}{k} = n\binom{n-1}{k-1}$
      • Story proof:
        • Picking $k$ Cabinet members out of $n$ people, with one selected as the Prime Minister
        • the same as picking one person from $n$ people as the Prime Minister, the rest as $k-1$ the Cabinet members
    • Vandermonde’s identity $\binom{m+n}{k} = \sum_{j=0}^{k}\binom{m}{j}\binom{n}{k-j}$
      • Story proof:
        • Picking $k$ people from $m+n$ places = the same as picking $j$ people from place $m$, $k-j$ people from place $n$

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Ref

• Alexandre-Théophile Vandermonde

Thoughts

• Labeling people is dangerous, but labeling events in a probability problem is important
• Think about the subtle differences between probabilities of
  • choosing one people in a team from four people
    • $P = \binom{4}{1}$
  • choosing three people in a team from four people
    • $P = \binom{4}{3}$
  • choosing one people in a team, three people in another team from a total of four people
    • $P = \binom{4}{1} = \binom{4}{3}$
  • choosing two people in a team, two people in another team from a total of four people
    • $P = \frac{\binom{4}{2}}{2}$

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