The Balls in the
Barrel Problem
DavidBedford
BallsinBarrels
Abarrelcontainsinfinitelymanyballsnumbered1,2,3,4,…
Thehourfrom11pmtomidnightisdividedintoinfinitely
manyintervalsT1,T2,T3,…
BallsinBarrels
T1:takeoutballsnumber1and2
T2:replaceballnumber1
T3:takeoutballsnumber3and4
T4:replaceballnumber3
T5:takeoutballsnumber5and6
T6:replaceballnumber5andsoon…
At1minutepastmidnight…
Howmanyballsareoutsidethebarrel?
Alltheoddnumberedballsareinthebarreland
alloftheevennumberedballsareoutside.
So,aswe’dexpect(?),thereareinfinitely
manyballsoutsidethebarrel.
Let’stryagainwithadifferentstrategy
Onceagainthehourfrom11pmtomidnightisdivided
intoinfinitelymanyintervalsT1,T2,T3,…
T1:takeoutballsnumber1and2
T2:replaceballnumber1
T3:takeoutballsnumber3and4
T4:replaceballnumber2
T5:takeoutballsnumber5and6
T6:replaceballnumber3andsoon…
At1minutepastmidnight…
Howmanyballsareoutsidethebarrel?
ThisQmeeveryballistakenoutonceANDevery
ballisreplacedonce.
ThisQmetherearenoballsoutsidethebarrel.
Let’stryagainwithadifferentstrategy
Onceagainthehourfrom11pmtomidnightisdivided
intoinfinitelymanyintervalsT1,T2,T3,…
T1:taketwoballsout
T2:putoneballback
T3:taketwoballsout
T4:putoneballback
T 5:taketwoballsout
T6:putoneballbackandsoon…
At1minutepastmidnight…
Howmanyballsareoutsidethebarrel?
Howmanywouldyoulike?
Let’stryagainwithnumberedballs
ThisQmewechoosetheballtobereplaceduniformlyatrandom
T1:takeoutballsnumber1and2
T2:pickoneatrandomandreplace
T3:takeoutballsnumber3and4
T4:pickone(outof3)atrandomandreplace
T5:takeoutballsnumber5and6
T6:pickone(outof4)atrandomand
replaceandsoon…
At1minutepastmidnight…
Howmanyballsareoutsidethebarrel?
Obviously0or
1
Theyallendupbackinthebarrel.