Introduction to Probability

Probability quantifies uncertainty. It is a measure of “how likely” an “event” can occur. Probability is measured on a scale of 0 to 1, both inclusive. An event that can never occur has probability 0 and an event that is certain to occur has a probability 1.

Some Basic Terminologies

  • Sample Space: The sample space Ω, is the set of possible outcomes of an experiment.
  • Sample Outcome: Points ω in Ω are called Sample Outcomes or realizations.
  • Events: Events are subsets of Ω

Example: Let us suppose that the experiment is tossing a coin twice.
Sample Space: {HH, HT, TH, TT}
Sample Outcome: HH, HT, TH, TT (each point in Sample space)
Events: first toss is head A = {HT, HT}, both tosses are different B = {TH, HT}

  • |A| : Number of points in A (if A is finite)
  • Ac : A complement (not A)
  • A ∪ B: A union B (A or B)
  • A ∩ B or AB: A intersection B (A and B)
  • A – B : set difference (points in A that are not in B)
  • A⊂B : set inclusion (A is a subset of or equal to B)
  • Φ : Null event (always false)
  • Ω : True event (always true)

Disjoint or Mutually exclusive events

We say that A1, A2, … are disjoint or are mutually exclusive if Ai∩Aj = φ whenever i ≠ j.

Now let’s get some intuitions on this definition. Disjoint/ Mutually exclusive events are those events whose probability of occurring together is zero. The probability of occurring of either of them is equal to the sum of probability of each occurring.

Example: Let us suppose a dice is rolled.
Events: A = {4, 6}, B = {3, 5}
You can notice that the probability of occurring together is zero i.e. A ∩ B is Φ.
P(A) = 1/3, P(B) = 1/3 and P(A ∪ B) = 2/3 which is equal to P(A) + P(B) which implies that A and B are disjoint.

What if the events are not mutually exclusive. Then there will be some overlap of both the events. So when probability will be added, the overlap will be added twice. Therefore, formula P(A ∪ B) becomes
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
which is basically deduction of overlap of both the sets.


Probability

We assign a real number P(A) to every event A called Probability of A.
P is called Probability distribution or probability measure.

A function P that assigns a real number P(A) to each event A is a probability distributionor a probability measure if it satisfies the following three axioms:
Axiom 1: P(A) ≥ 0 for every A.
Axiom 2: P(Ω) = 1
Axiom 3: If A1, A2,… are disjoint then
Probability

There first two axioms are quite intuitive, whereas the third is the generalized formula of disjoint/mutually exclusive events. The Ai in axiom 3 can be considered as a partition of event A summarizing the formula as
P(A) = P(A1) + P(A2) + P(A3) + . . .

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