__Elementary Probability Ideas____What has a Probability?__

A person has a question in mind:

That asks about a proposition (statement, assertion,
conjecture, etc.):

The person wants to know: What is the probability that this
proposition is true?

This asks about an event (something of a certain sort
happening).

The person wants to know:What is the probability of this
event occurring?

Obviously these are two different ways of asking the same
question.

__Propositions and Events__

Logicians are interested in arguments from premises to
conclusions. Premises and conclusions are propositions. So inductive logic
textbooks usually talk about the probability of propositions.

Most statisticians and most textbooks of probability talk
about the probability of events.

So there are two languages of probability, propositions and
events.

Propositions are true or false.

Events occur or do not occur.

__Notation:Logic__

Propositions or events are represented by capital letters:A,
B, C…

Logical Compounds
will be represented as follows, no matter whether we have propositions or
events in mind:

Disjunction (or): AvB for (A, or B, or both). We read this
“A or B.”

Conjunction (and): A &B for (A and B).

Negation (not)~A for (not A).

__Notation:Sets__

Statisticians usually do not talk about propositions. They
talk about events in terms of set theory. Here is a rough translation of
proposition language into event language.

The disjunction of two propositions, AvB, corresponds to the
union of two sets of events.

The conjunction of two propositions, A&B, corresponds to
the intersection of two sets of events.

The negation of a proposition, corresponds to the complement
of a set of events.

__Two Conventions__

Probabilities lie between 0 and 1.

The probability of a proposition that is certainly true, or
of an event that is sure to happen, is 1.

__Mutually Exclusive__

Two propositions are called mutually exclusive if they can’t
both be true at once.

Likewise, two events which cannot both occur at once are
called mutually exclusive. They are also called disjoint.

__Adding Probabilities__

The probabilities of mutually exclusive propositions or
events add up.

Adding probabilities is for mutually exclusive events or
propositions.

__Independence__

Intuitively:

Two events are independent when the occurrence of one does
not influence the probability of the occurrence of the other.

Two propositions are independent when the truth of one does
not make the truth of the other any more or less probable.

Multiplying probabilities is for independent events or
propositions.

__Compounding__

Throwing a 6 with one die is a single event. Throwing a sum
of seve with two dice is a compound event. It involves two distinct outcomes,
which are combined in the event “the sum of the dice equals 7.”

__Laplace__

The below example was a great favourite with P.S.de
Laplace(1749-1827).

In an experiment once you have picked an urn with a bias for
a given colour, it is more probable that both balls will be of that colour,
than that you will get one majority and one minority colour.

Reference-Hacking, Ian; An Introduction to Probability and
Inductive Logic, 2001, CUP.

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