Review

Probability

History
  • Gambling, e.g. Girolamo Cardano on dice in 1560, Abraham de Moivre “Doctrine of Chances” in 1718
  • (Finite) set of outcomes, play the game, one outcome happens
  • Players sometimes realized their intuition about something was wrong and they’d consistently lose money. So they tried to do rigorous calculations to figure out why their intuition was off.
A diversion into set theory
  • Set \(S\) is called sample space or “universe” (sometimes use \(\Omega\))
  • Examples: 6-sided die: \(\{ 1, 2, 3, 4, 5, 6 \}\), coin toss: { H, T }, twitter: { No action, Reply, Retweet, Like }, the winner of an election is a { Democrat, Republican, Independent, Libertarian, Green, Socialist, actual cartoon character }, etc.
  • What are the outcomes when tossing a coin 3 times? (Write on board, don’t erase, e.g. HHH, HHT, …)

  • To say the outcome \(s\) is in the set \(S\) (also called a member or element of \(S\)), we write \(s \in S\).
  • Events: subsets \(E \subseteq S\) are called events.
  • Examples: roll a 6: \(\{ 6 \}\), roll even: \(\{ 2, 4, 6 \}\), the empty set \(\emptyset\) etc.
  • The event \(E\) “happens” if the outcome of the game \(s \in E\).
  • Set operations: the union \(E_1 \cup E_2\) contains outcomes \(s\) if \(s \in E_1\) or \(s \in E_2\) (or both). e.g. \(\{ 1, 2 \} \cup \{ 4 \} = \{ 1, 2, 4 \}\)
  • Set operations: the intersection \(E_1 \cap E_2\) contains outcomes \(s\) if \(s \in E_1\) and \(s \in E_2\). e.g. \(\{1, 2 \} \cap \{ 2, 4 \} = \{ 2 \}\).
  • Disjoint events: \(E_1\) and \(E_2\) are disjoint if \(E_1 \cap E_2 = \emptyset\), i.e. there is no overlap.
  • Set operations: the complement \(E^c\) of an event \(E\) is all outcomes that are not in \(E\). e.g. for the 6-sided dice: \(\{ 2, 4, 6 \}^c = \{ 1, 3, 5 \}\).
  • Set operations: the size of a set \(|E|\) is the number of outcomes it contains. e.g. \(|\emptyset| = 0\).
  • What is \(|S|\) for tossing a coin \(n\) times?

Probability
  • Probability: measure for events, \(P(E)\) is the likelihood of \(E\) happening
  • Rules: \(P(S) = 1\), \(P(\emptyset) = 0\), for any E, \(0 \leq P(E) \leq 1\)
  • Complement event: \(P(E^c) = 1 - P(E)\) (sometimes easier to calculate one or othe other)
  • If I toss coin 3 times, what’s probability of getting at least one H?
  • What if I toss a coin \(n\) times?
  • Equiprobable outcomes: for every \(s \in S\), \(P(s) = 1/|S|\). Reasonable for (fair!) games, maybe not for some of our other examples…
  • Counting rule: with equiprobable outcomes \(P(E) = |E|/|S|\).
  • Count the number of ways \(E\) can happen, divide by total number of things that could happen.
  • Addition rule: for disjoint events \(E_1, E_2\), \(P(E_1 \cup E_2) = P(E_1) + P(E_2)\).
  • Addition rule: for any events, \(P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2)\) (subtract off the double-counted overlap).

  • Conditional probability: \(P(E_2 | E_1) = P(E_2 \cap E_1)/P(E_1)\) (draw Venn diagram)
  • Intuition: given that \(E_1\) happened, make \(E_1\) the new sample space
  • e.g. What if \(E_1\) is { Democrat, Republican } and \(E_2\) is { Democrat }?