Let us summarize what concepts we've encountered up to now in one form or the other. Note: this summary is exceptionally important. Make sure that you have a good idea about what all’s involved.
The result of a random experiment . For example, in the tossing of a coin, obtaining “Heads” is a possible outcome.
(ii) Equally likely
Two outcomes and of an experiment can be said to be equally likely when there is no evident reason to favor over or viceversa. To make the idea more concrete, you can say that as you repeat the experiment an indefinitely large number of times, the relative occurrence of and will be equal. For example, if you toss a fair coin an indefinitely large number of times, the relative occurrence of both Heads and Tails will be . Similarly, if a die is rolled an indefinitely large number of times, each of the six faces will have a relative occurrence of .
An event is a set of outcomes. Thus, an event can be viewed as a subset of the universe of all outcomes of the experiment, which is termed as the sample space of the experiment.
For example, in drawing a card from a well-shuffled deck of cards at random, the sample space is of size . The event defined as
: The card drawn is red is a set of outcomes.
The event defined as
: The card drawn is a king
is a set of outcomes.
The event defined as
: The card drawn is the Ace of spades
is a set of only outcome, and is thus an elementary event, whereas and are compound events.
If all outcomes in an experiment are equally likely (like in tossing a fair coin, rolling a fair die, drawing a card at random from a well-shuffled deck), then the probability of occurrence of an event is simply
(Note also that probability defined this way is not applicable to situations like weather prediction, which use advanced probabilistic models).
(v) Events as sets
Since events are sets of outcomes, set operations can be defined for events. Let and be two events associated with a random experiment. In general, we have
In case of mutually exclusive events, this reduces to
(vi) Conditional Probability
If and are two events, then the probability of occurring given that has already occurred is termed the conditional probability of given , and is written as . We have,
(vii) Independent events
If and are independent events, then
which implies that
This basically means that the probability of occurring is not affected by the occurrence or non-occurrence of , and vice-versa.