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.
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).
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.
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.
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.
The probabilities of mutually exclusive propositions or events add up.
Adding probabilities is for mutually exclusive events or propositions.
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.
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
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.