Review

Probability

Conditional probability
  • Important for at least two reasons: (1) can calculate probabilities for complicated events by breaking it into steps, and (2) gives a way to adjust probabilities based on new information
  • Intuitively, if you know \(E_1\) has occurred, make \(S = E_1\) the new sample space
  • Conditional probability: \(P(E_2 | E_1) = P(E_2 \cap E_1)/P(E_1)\). This tells us how probability in the new sample space relates to probability in the original one
    1. Multiplication rule: \(P(E_2 \cap E_1) = P(E_1)P(E_2 | E_1)\) “probability that two things both happen”
  • Deck of 52 cards, 4 are aces. What is \(P(E)\) for \(E =\) { first two cards are aces }.
  • \(E_1 =\) { first card is an ace }, \(E_2 =\) { second card is an ace }, 4/52, then new sample space has 51 cards, 3 of them aces (about 0.45%)
  • mean(replicate(10000, all(sample(1:52, 2) <= 4)))
Independence
  • One of the most important concepts in statistics. An underlying assumption of many common statistical methods. Something to always keep in mind…
  • Independence: \(E_1\) and \(E_2\) are independent if \(P(E_2 | E_1) = P(E_2)\)
  • Knowing that one has happened doesn’t change the probability of the other one
  • If I toss a coin 10 times, what is the probability the 10th toss is H given the first 9 are T?
  • Otherwise, they are called dependent
  • (In)dependence can make or break the accuracy/success of a probability model. Common error to believe/assume things are independent when they aren’t
  • e.g. 2016 election: predictions that showed Secy. Clinton with a high chance of winning relied on independence
  • e.g. Financial crisis: more houses foreclosing in a neighborhood lowered property values, increasing probability of mortgage problems for other houses in the same neighborhood

  • Multiplication rule: for independent events, \(P(E_1 \cap E_2) = P(E_1)P(E_2)\)
  • This works for more than 2 events: e.g. probability of all \(n\) coin tosses being H multiply (1/2) \(n\) times, get \(1/2^n\)

  • Drawing with and without replacement (check out the sample() function in R)
  • Deck of cards with replacement: draw one card, record it, return it, shuffle, draw a second card
  • Deck of cards without replacement: draw one card, record it, set it aside, draw another from the remaining cards

  • Important: which version of the multiplication rule should you use? Depends on (in)dependence
  • Important: multiplication rule is for both events, addition rule is for either event
  • Venn diagrams

Bayes’ theorem
  • Suppose some facial recognition software attempts to identify people with an outstanding arrest warrant from a database of photos. It inputs a photo of a person and classifies as a match to the database, or not a match. If the photo input is someone does not have a warrant, there is a 1% chance it will mistakenly match them anyway, but it is 100% accurate otherwise.
  • This software is given camera feeds from all over the city
  • There is a match. What’s the probability the match is accurate?
  • Let \(M\) denote a declared match, and \(W\) denote that the person actually has an outstanding warrant. We know \(P(M | W) = 1\), and \(P(M | W^c) = .99\)
  • Do we have enough information?