Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CHAPTER 9: INTRODUCING PROBABILITY Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. Flip a coin. What is the probability that the coin comes up heads? o Classical probability – assume all outcomes are equally likely o Count Buffon (1707-1788) 2048/4040 = .5069 o Karl Pearson (1900) 12,012/24,000 = .5005 o John Kerrich 5067/10,000 = .5067 Random – individual outcomes are uncertain but there is a nonetheless regular distribution of outcomes in large numbers of repetitions Probability – proportion of times the outcome would occur in a very long series of repetitions o Trials must be independent o Probability is empirical o Computer simulations Probability models o Sample space (S) – set of all possible outcomes o Event (E) is an outcome or set of outcomes (subset of S) o Probability model consists of a sample space and a way of assigning probabilities to events Probability Rules o 0 ≤ P(E) ≤ 1 o Sum of all probabilities = 1 o Complement Rule: P(E) + P(E’) = 1 o Addition Rule for Disjoint Events P(A or B) = P(A) + P(B) Example: Two fair dice are rolled. Calculate the following probabilities. 1. Doubles are rolled 2. A sum of six is rolled 3. A sum of seven or nine is rolled 4. The sum of the digits is odd Normal Probability Models o Normal distributions are probability models o Example: Suppose the heights of American women are distributed N(64, 2.7). What proportion of women are more than 68 inches tall? What is the probability that a randomly chosen woman is more than 68 inches tall? Random Variables o Variable whose value is a numerical outcome of a random phenomenon o Can be discrete or continuous Probability Distribution o Tells what values a random variable X can take and how to assign probabilities to those values Personal Probability – expresses an individual’s judgment of how likely an outcome is