Random Thoughts 2012 (COMP 066)

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

2. 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 2008. Use the Bayes rule to compute the probability. • Read in the Moldinov book chapter 6.

3. Pregnancy rate by age group

4. Bayes Rule • Bayes rule

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

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

7. 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 = pn • 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

8. 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

9. Expected Value Σ • 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 + 0.1 * 5 + 0.4 * 1 = $-0.10

10. 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

11. Odds vs. Probability • Odds vs Probability

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

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

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

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