Random Thoughts 2012 (COMP 066)

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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 2008. Use the Bayes rule to compute the probability.

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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 2008. Use the Bayes rule to compute the probability.
• Read in the Moldinov book chapter 6.
Bayes Rule
• Bayes rule
Bayes Rule for Pregnancy Test

Age 27: [99.99%, 84.16%]

Age 44: [ 99.96%, 37.67%]

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
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
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
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
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
Odds vs. Probability
• Odds vs Probability
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)
Description of Data
• Mean
• Average
• Median
• Middle value
• Standard deviation
• Variability or spread of the data
• Percentile
• Position within ordered list of values
Confidence Interval
• Margin of error of N samples

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