Preface
If you tos s a fai r coi n man y times , yo u woul d expec t th e coi n t o lan d
heads as often a s tails. Th e goa l of this boo k i s to mak e thi s intuitio n
precise. A s th e numbe r o f tosse s increases , th e proportio n o f head s
approaches 1/2, bu t i n wha t way , ho w quickly , an d wha t deviation s
should w e expect ? Heads or Tails i s a n introductio n t o probabilit y
theory; i n particular, i t i s an introductio n t o the stud y o f convergenc e
properties o f sequence s o f observations . I n thi s book , I wil l presen t
an are a o f mathematic s tha t ha s bot h utilit y an d beauty .
Probability theor y i s th e branc h o f mathematic s concerne d wit h
the stud y o f rando m phenomena . A random phenomenon i s a n ex -
periment wit h a n outcom e tha t depend s o n chance , eithe r becaus e
the exac t condition s fo r it s outcom e ar e no t know n o r becaus e th e
randomness o f th e experimen t actuall y exists . However , w e wil l no t
discuss th e source s o f randomnes s i n rando m phenomena ; instead ,
we wil l star t wit h a mathematica l mode l o f probability . Heads or
Tails present s a n introductio n t o th e mathematica l model s o f thes e
phenomena an d t o th e rigorou s deductio n o f th e law s w e expec t th e
outcomes o f sequence s o f independen t experiment s t o follow .
While writing this book, I kept th e following thre e points in mind.
1. A freshman - o r sophomore-leve l analysi s cours e i s al l tha t
is neede d t o understan d th e materia l i n thi s book . I n particular ,
a knowledg e o f measur e theor y i s no t necessary . Thi s boo k i s aime d
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