STAT 211 Prof Parzen CHAPTER 2
PROBABILITY, CONDITIONAL PROBABILITY, BAYES
Probability theory enables us to measure
uncertainity,
chance, likelihood.
Probability theory has applications to ex
plain and predict observations in every aspect of life: science,
engineering, management, financial planning for retirement,.
Statistics learns from data
probability models
that fit observed
data (by computing probabilities that a model is true).
We study
probability theory because it is the basis for
statistical applica
tions.
2.0 COIN TOSSING
D
ecisions are often made by tossing a coin which has two possi
ble outcomes;
head (often denoted as 1 or success S or s) and
tails (often denoted 0 or failure F or f).
When observing the sex
(gender) of a newly born baby: possible outcomes are male (of
ten denoted 4) and female (often denoted 6)
The outcome of a single coin toss (the gender of a new baby) is
unpredictable.
However one may be able to predict proportions
(of
a large number of observations with
specified outcomes).
DEEFINITION: Denote by Proportion [heads] the proportion of
heads in a large number of coin tosses; it
is a number (between
0 and 1). Experience leads us to believe proportion is reproduci
ble approximately and what is
predictable are intervals in
which the observed proportion lies with high probability.
The
concept of probability of heads on a coin toss, denoted
Prob [heads],
is defined on order to provide a model to predict
and/or
explain
the observed proportion of heads in a large number of coin
tosses.
PROBABILITY AND STATISTICAL
INFERENCE: Prob
ability makes assumptions about probability p of heads in a coin
toss, and computes probabilities that future values of Proportion
[heads] will be in a specified interval.
Statistical inference ob
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serves the value of Proportion
[heads], in n tosses of a coin
whose probability p is not assumed known, and determines
probabilities that
p is in a specified interval.
Fair coin
:
A coin whose probability
p of heads equals .5 is
called a fair coin.
We apply this concept in practice by
expect
ing a fair coin to fall heads approximately 5000 times in 10000
tosses (50 heads in 100 tosses),
Proportion
Heads in 10000 trials Fair Coin
=5000 approximately
Precise statements are not made about the exact number of
heads. Rather we say with probability 95% a fair coin will fall
heads in
10000
n
=
tosses
a proportion between .49 and .51;
in
100
n
=
tosses of a fair coin, the proportion of heads will be
between .4 and .6 with probability 95%.
Note that the interval of proportions we expect to observe
changes as we change the sample size
n
, and converges to a sin
gle number (the true value p) as the sample size tends to infinity.
2.1 SAMPLE SPACE, EVENTS, EVENT OPERATIONS
To record (or to make a prediction about)
the outcome of an
observation (measurement or experiment) we usually decide in
advance the set (denoted S) of all outcomes that could be ob
served;
the sample space
S
is a set or
list of descriptions of the
samples or outcomes.
Any data set that we could observe
by
our measurements is decribed by a member of S.
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 Fall '07
 Parzen
 Conditional Probability, Probability, Probability theory

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