06 Bernoulli Trials
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!" $%&'() $'%*) %+, -./0%01(123
-%.2 456 -%7*%( 8.1%+9() %+,
:).+/;((1 8.1%(7
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=/++)*G/+ 0)2H))+ -%7*%( 2.1%+9() %+,
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• K !"#$%&''( *#(+' 17 %+3 './0%01(17G* )L').1&)+2 H12D 2H/
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=/++)*G/+ 0)2H))+ -%7*%( 2.1%+9() %+,
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–
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– M1(( O/H Q/+)7 9/ ;' 2/&/../HN
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=/++)*G/+ 0)2H))+ -%7*%( 2.1%+9() %+,
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-./0%01(123 /E - 7;**)77)7 1+ $
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-./0%01(123 /E - 7;**)77)7 1+ $
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-./0%01(123 /E - 7;**)77)7 1+ $
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-./0%01(123 /E - 7;**)77)7 1+ $
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-./0%01(123 /E - 7;**)77)7 1+ $
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K+ 1+2).)7G+9 './').23 /E 01+/&1%(
*/)_*1)+27
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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:1+/&1%( './0%01(1G)76 1((;72.%G/+
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:1+/&1%( './0%01(1G)76 1((;72.%G/+
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=/&&)+27 /+ 01+/&1%(
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=/&&)+27 /+ 01+/&1%(
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=/&&)+27 /+ 01+/&1%(
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(%.9) $ 7D/H+ %0/B) 17 % &%+1E)72%G/+ /E 2D)
=`8"
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4+2).)7G+9(3C H) 9)2 2D) 0)(( *;.B) )B)+ E/.
%73&&)2.1* 01+/&1%( './0%01(1G)7
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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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If p=0.8, when estimating based on 1000 flips,
it’s extremely unlikely to make a mistake ofmore than 0.05.
•
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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If p=0.8, when estimating based on 1000 flips,
it’s extremely unlikely to make a mistake ofmore than 0.05.
•
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.
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8D) @??R c.%+F)+Z=/()&%+ )()*G/+
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•
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-./0%01(1G)7 E/. E.%*G/+7 /E c.%+F)+ B/2) 1+ '.)
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c.%+F)+ %+, =/()&%+ B/2)7 */&01+),X
• 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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