Random Thoughts 2012 (COMP 066) Jan-Michael Frahm Jared Heinly

Assignment Calculate the probability of being pregnant with a positive pregnancy test for a women with age 27 and for a women of age 44 in Use the Bayes rule to compute the probability. Read in the Moldinov book chapter 6. 2

Pregnancy rate by age group 3

Bayes Rule Bayes rule 4

Bayes Rule for Pregnancy Test Age 27: [99.99%, 84.16%] Age 44: [ 99.96%, 37.67%] 5

Spam filtering Often done based on black list  too restrictive  easy to evade by putting false sender Bayes rule can be used to perform spam filtering Filtering based on words in the  “viagra” has high probability of spam  “Bayes-rule” has low probability of spam can be learned from s 6

Probability Rules Probability of event = p  ex. probability of rolling a 1 on a die: p = 1/6 Probability of event not happening = 1 – p  ex. probability of not rolling a 1: p = 5/6 Probability of event happening n times in a row = p n  ex. probability of rolling five 1s in a row: p = (1/6) 5 Probability of event happening at least once during n attempts = Inverse of probability of event not happening n times in a row = 1 – (1 – p) n  ex. probability of rolling a 1 at least once in 5 rolls: p = 1 – (5/6) 5 7

Probability Rules Probability of event happening k times in n attempts  Binomial Can only add probabilities when you want to know if any one of a set of outcomes occurred and it is impossible for the outcomes to occur at the same time  ex. probability of rolling a 1 or a 2 on a die: p = 2/6 8

Expected Value Expected value = probability of event * value of event Ex: pay $1 to play a game, 10% chance of winning $5, 40% chance of winning $1 Expected Value = * * 1 = $ Σ

Perceptual Pitfalls The probability that two events will occur can never be greater than the probability that each will occur individually.  “a good story is often less probable than a less satisfying … [explanation]” Missing information Availability bias  recallable prior knowledge influences our estimates 10

Odds vs. Probability  Odds vs Probability 11

Binomial distribution Binomial distribution: For events with K successes in N trials Properties of a Binomial distribution: 1)Fixed number of trials 2)Only outcomes are success and fail? 3)Same probability for success in each trial 4)Independent trials (no influence of previous trials to current trial) 12

Description of Data Mean  Average Median  Middle value Standard deviation  Variability or spread of the data Percentile  Position within ordered list of values 13

Confidence Interval Margin of error of N samples 14 z*=z*= Number of samples needed:

How many trials? Margin of error for a population proportion  Depends on proportion in the population that had the characteristic we searched for 15