06_Bernoulli_Trials_packed

Pollak ilya pollak ilya pollak ilya pollak ilya

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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 two-way 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 p-0.5=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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