1
Chapter 5: Probability
Read Chapter 5
Random Phenomena
For random phenomena, the outcome is
uncertain
In the shortrun, the proportion of times that
something happens is highly random
In the longrun, the proportion of times that
something happens becomes very predictable
Probability quantifies longrun randomness
Law of Large Numbers
As the number of trials increases, the proportion
of occurrences of any given outcome
approaches a particular number “in the long run”
For example, as one tosses a coin, in the long
run 1/2 of the observations will be a Head.
Coin Flipping Applet
Probability
With random phenomena, the
probability
of a particular outcome is the proportion of
times that the outcome would occur in a
long run of observations
Example:
When rolling a die, the outcome of “6” has
probability = 1/6.
In other words, the
proportion of times that a 6 would occur in a
long run of observations is 1/6.
Independent Trials
Different trials of a random phenomenon
are
independent
if the outcome of any one
trial is not affected by the outcome of any
other trial.
Example:
If you have 20 flips of a coin in a row that are
“heads”, you are not “due” a “tail”  the
probability of a tail on your next flip is still 1/2.
The trial of flipping
a coin is independent of
previous flips.
How do we calculate Probabilities?
Calculate theoretical probabilities based on
assumptions about the random phenomena
.
For
example, it is often reasonable to assume that
outcomes are equally likely such as when
flipping a coin, or a rolling a die.
Observe many trials of the random phenomenon
and use the sample proportion of the number of
times the outcome occurs as its probability. This
is merely an estimate of the actual probability.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
Looking out for #1: The first digit (Benford
’
s)
law
A phenomenon on the first digit
of numbers.
Numbers start with smaller digit
rather than bigger digits
In a logarithm book,
Simon Newcomb
noticed
the pages started dirty and got cleaner toward
the back
Numbers more frequently begin with a ‘1’ than
any other number, and the frequency decreases
up to nine
Scope of First Digit Law
Census statistics
Addresses
Stock market
Accounting figures
Newspapers
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 VELLEMANP
 Probability, Probability theory, Flipism, Coin flipping, Frank Benford

Click to edit the document details