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Unformatted text preview: ya Pollak Ilya Pollak Ilya Pollak Ilya Pollak Ilya Pollak Ilya Pollak This tells us how to empirically esGmate the probability of an event! • To esGmate the probability p based on n ﬂips, divide the observed number of H’s by the total number of experiments: k/n. • To see the distribuGon of k/n for any n, simply rescale the x axis in the distribuGon of k. • This distribuGon will tell us – What we should expect our esGmate to be, on average, and – What error we should expect to make, on average Ilya Pollak Ilya Pollak Ilya Pollak Ilya Pollak Ilya Pollak Note:
o for 50 flips, the most likely outcome is the correct one, 0.8
o it’s also close to the “average” outcome
o it’s very unlikely to make a mistake of more than 0.2 Ilya Pollak Ilya Pollak Ilya Pollak If p=0.8, when estimating based on 1000 flips,
it’s extremely unlikely to make a mistake of
more than 0.05.
• Hence, when the goal is to forecast a twoway
election, and the actual p is reasonably far from
1/2, polling a few hundred people is very likely
to give accurate results.
• However,
o independence is important;
o getting a representative sample is important
(for a country with 300M population, this is
tricky!)
o when the actual p is extremely close to 1/2
(e.g., the 2000 presidential election in Florida or
the 2008 senatorial election in Minnesota),
pollsters’ forecasts are about as accurate as a
random guess. Ilya Pollak The 2008 Franken Coleman elecGon • Franken 1,212,629 votes • Coleman 1,212,317 votes • In our analysis, we will disregard third party candidate who got 437,505 votes (he actually makes pre elecGon polling even more complicated) • EﬀecGvely, p ≈ 0.500064 Ilya Pollak ProbabiliGes for fracGons of Franken vote in pre
 elecGon polling based on n=2.5M (more than all Franken and Coleman votes combined) • Even though we are unlikely to make
an error of more than 0.001, this is not
enough because p0.5=0.000064!
• Note: 42% of the area under the bell
curve is to the left of 1/2.
• When the election is this close, no poll
can accurately predict the outcome.
• In fact, the noise in the voting process
itself (voting machine malfunctions,
human errors, etc) becomes very
important in determining the outcome. Ilya Pollak EsGmaGng the probability of success in a Bernoulli trial: summary • As the number n of independent experiments increases, the empirical fracGon of occurrences of success becomes close to the actual probability of success, p. • The error goes down proporGonately to n1/2. I.e., error aler 400 trials is twice as small as aler 100 trials. • This is called the law of large numbers. • This result will be precisely described later in the course. Ilya Pollak...
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 Fall '08
 GELFAND
 Electrical Engineering

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