1
Probability
1.1 Introduction
Bayesian thinking is based on the subjective viewpoint of probability. In this
chapter, we will talk about the different ways of thinking about probability.
1.2 Measuring Uncertainty
We live in a world of uncertain events and some events are more likely to
occur than other events. We use words such as “likely”, “probable”, “possible”,
“rare”, and “maybe” to describe this uncertainty. It is natural to use numbers
that we call probabilities to quantify this uncertainty. The probability of an
event
A
, denoted
P
(
A
), is a number between 0 and 1 assigned to the event
A
where a larger number indicates that the event is more likely to occur.
Some uncertain events already have numbers assigned to them. In games
of chance where dice are rolled or cards are dealt from a wellshuffled deck,
outcomes have particular probabilities. For example, the chance of rolling two
dice equal to doublesixes is 1/36 and the chance of dealing a four of diamonds
in a regular deck is 1/52. In actuarial tables, there are assigned probabilities
that a person’s life span will be a particular length based on one’s gender and
age. These actuarial tables are used by insurance companies to write up a life
insurance policy and decide on the cost of the policy to the customer.
There are two ways of viewing probabilities that allow us to assign proba
bilities in games of chance and actuarial tables. The classical or“equallylikely”
view assumes that one can represent the outcomes of a random experiment in
such a way such that the outcomes are equally likely. Then each outcome is
assigned a probability equal to one divided by the total number of outcomes.
In the dice example, there are 36 equally likely ways of representing the pos
sible rolls of the dice, and the probability of one outcome (two sixes) is equal
1/36. In the card example, there are 52 possible draws in a deck of cards,
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2
1 Probability
and if the deck is wellshuffled, each outcome such as “four of diamonds” is
assigned the probability of 1/52.
A second way of thinking about probabilities is based on longrun relative
frequencies. Suppose you are able to repeat a random experiment many times
under similar conditions. Then the probability of an event is approximated
by its relative frequency in the large number of trials. This viewpoint can be
applied in games of chance. For example, the probability that the sum of two
dice is equal to 7 can be approximated by the relative frequency of 7 in many
rolls of the two dice. This definition can also be used for actuarial tables. The
chance that a male of age 70 will survive ten years can be approximated by
the relative frequency of 70year old males who survive ten years.
Is it possible use the relative frequency viewpoint to measure uncertainty
for all random events? Lindley makes a distinction between two types of
events.
Statistical events
are the events that can be repeated under similar
conditions and
nonstatistical events
are events that are essentially unique
and can not be repeated. Games of chance are statistical events, while one
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 Spring '10
 albert
 Probability, Probability theory, Proposition, Bayesian probability, Phillies

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