# Various Terminologies in probability

**1. Experiment: **

An experiment (strictly speaking, a random experiment) is an operation which can result in two or more ways. The possible results of the experiment are known as outcomes of the experiment. These outcomes are known in advance. We cannot predict which of these possible outcomes will appear when the experiment is done.

**2. Sample Space:**

The set containing all the possible outcomes of an experiment as its element is known as sample space. It is usually denoted by S.

**3. Event:**

An event is a subset of the sample space S.

**Illustration 1: **

Let us consider the experiment of tossing a coin. This experiment has two possible outcomes: heads (H) or tails (T)

Ã° sample space ( S) = {H, T}

We can define one or more events based on this experiment. Let us define the two events A and B as:

It is easily seen that set A (corresponding to event A) contains outcomes that are **favorable to event A and set B contains outcomes favorable to event B.**

Recalling that n (A) represents the number of elements in set A, we can observe that

n (A) = number of outcomes favorable to event A

n (B) =number of outcomes favorable to event B

n (S) = number of possible outcomes

Here, in this example, n (A) = 1 , n (B) = 1 , and n (S) =2.

# Probability of occurrence of an event

If S is the sample space of an experiment, then the probability of occurrence of an event A is defined as:-

Probability of occurrence of A =

Or, in other words, P (A) =

Note

Note

- In the example on tossing of a coin, the probabilities of occurrence of events A and B are:

â€‹ P (A) = = and P (B) = = - It is seen that P(S) = 1.
- The event, whose probability of occurrence is zero, is known as an
**Impossible Event**and the event, whose probability is 1, is known as a**Certain Event.** - In general for event A, 0 P (A) 1

# Odds in Favor and Odds against an Event

If, in an experiment, the number of outcomes favorable to event A is m and number of outcomes not favorable to event A is n, then:

**(i) **Odds in favor of A = =

**(ii) **Odds against A = =

**Illustration 2:**

Odds in favor of getting a spade when a card is drawn from a well shuffled pack of 52 cards are

= =

**Notes:**

If odds in favor of an event are a: b, then the probability of the occurrence of that event is a/ (a + b) and the probability of the non-occurrence of that event is b/ (a - b).

# Axiomatic and other approaches

Neither the Bayesian nor the frequentist approach really gives us a satisfactory formal structure for development of a rigid theory.

The axiomatic approach takes a different tack. Instead of focusing on the question "What is probability?" we step back and ask "How does probability *work*?"

We then focus on abstract mathematical concepts such as sets, measure, sigma algebras and a set of rules we expect probability to follow, known as Kolmogorov's axioms.

But the ultimate justification for this approach rests on experience, too. Fortunately, the results derived by applying these axioms accord very nicely with experience. We can also derive Bayes' theorem as a consequence, and we can show that in a large number of trials, the frequencies will give us a good estimate of the probabilities; or at least it will do so *on average*.

**Illustration 3:**

**Example of the axiomatic approach**

Given a standard deck of card. We want to draw a card at random.

This experiment can be modeled by a set of 52 possible outcomes. Any set with 52 elements has exactly subsets. To each of those subsets we may assign a certain number (a "probability"), so that certain axioms are satisfied. We choose to assign probability 0.25 to the subset consisting of all 13 hearts. We also assign 0.25 then the subset consisting of all spades.

According to the axioms, we must then assign probability 0.5 to the subset consisting of every spade and heart in the deck.

In non-axiomatic informal terms, we would describe this result thus:

If the probability that a card (drawn at random from a standard deck of cards) is a heart is .25, and the probability that it is a spade is also .25, and if I know that the being a heart and being a spade are mutually exclusive possibilities (i.e., a card cannot be both), then the probability that it is a heart *or* a spade is 0.5 = 0.25 + 0.25.

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