Random Thoughts 2012(COMP 066)

Jan-Michael FrahmJared Heinly

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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.

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Pregnancy rate by age group

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Bayes Rule• Bayes rule

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Bayes Rule for Pregnancy Test

Age 27: [99.99%, 84.16%]Age 44: [ 99.96%, 37.67%]

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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

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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

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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

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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 + 0.1 * 5 + 0.4 * 1 = $-0.10

Σ

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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

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Odds vs. Probability Odds vs Probability

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Binomial distribution• Binomial distribution: For events with K

successes in N trials

• Properties of a Binomial distribution:1) Fixed number of trials2) Only outcomes are success and fail?3) Same probability for success in each trial4) Independent trials (no influence of previous trials

to current trial)

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Description of Data• Mean

Average• Median

Middle value• Standard deviation

Variability or spread of the data• Percentile

Position within ordered list of values

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Confidence Interval• Margin of error of N samples

z*=

Number of samples needed:

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How many trials?• Margin of error for a population proportion

Depends on proportion in the population that had the characteristic we searched for