06 Bernoulli Trials

8/19/2019 06 Bernoulli Trials

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Since P(zero H's)+ P(one H) + P(two H's)+…+ P(n H's) =1,

it follows thatn

k

!

" #

$

% & pk (1' p)n'k =1.

k= 0

n

(Another way to show the same thing is to realize that

n

k

!

" #

$

% & pk

(1' p)n'k

=

( p+

(1' p))n=

1

n=

1.k= 0

n

(

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Note:

o

for 50 flips, the most likely outcome is the correct one, 0.8o

it’s also close to the “average” outcomeo

it’s very unlikely to make a mistake of more than 0.2

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If p=0.8, when estimating based on 1000 flips,

it’s extremely unlikely to make a mistake ofmore than 0.05.

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•

Hence, when the goal is to forecast a two-wayelection, and the actual p is reasonably far from

1/2, polling a few hundred people is very likely

to give accurate results.

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•

Hence, when the goal is to forecast a two-wayelection, 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


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• 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 bellcurve 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 veryimportant in determining the outcome.

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06 Bernoulli Trials