Probability

The probability of happening of an event A is

\( P(A) = \frac{\text{Number of favourable outcomes of event A}}{\text{Number of possible outcomes}} \)

Mathematical Probability

If a trial results in 'n' exhaustive, mutually exclusive and equally likely cases and 'm' of them are favourable to the happening of an event E then the probability of happening of E is given by

\( P(E) = \frac{\text{Favourable number of outcomes (cases)}}{\text{Exhaustive number of cases}} = \frac{m}{n} \)

Statistical (Empirical) probability

If a trial is repeated a number of times under essential homogeneous and identical conditions, then the value of the ratio of the number of times of the event happens to the number of all possible outcomes as the number of trials becomes indefinitely large is called probability of happening of the event.

A die is a well-balanced cube with its six faces marked with numbers (dots) from 1 to 6. It has a chance of happening events consisting of the single element is \(\frac{1}{6}\)

An unbiased coin has 50% chance of happening the event 'H' and 'T'. i.e. \(P(H) = \frac{1}{2} = P(T)\).

A pack of card consists of spade (13), Heart (13), Diamond (13) and club (13). So there is chance of happening spade, heart, diamond and club each equal to \( \frac{13}{52} = \frac{1}{4} \).

Equally likely events

Two events are said to be "Equally likely" if one of them can be expected in preference to other.
Probability of impossible events = 0 and probability of sure event = 1.

Mutually exclusive events

Two events A and B are said to be a mutually exclusive event, if \( A \cap B = \phi \). In other words, such events where the occurrence of one event precludes the occurrence of the other event.

Example:Let's consider a simultaneous toss of two coins, then sample space S = {HH, HT, TH, TT } ∴ \( A \cap B = \phi \).

Additive law for mutually exclusive events

If A and B are two mutually exclusive events, then \(P(A \cup B) = P(A) + P(B).\) If A, B and C are three mutually exclusive events then occurrence of at least one of these three events is \( P(A \cup B \cup C)= P(A) + P(B) + P(C). \)

Independent event: Two events are said to be independent if the occurrence of one does not depend upon the occurrence of other. E.g. on rolling a dice, tossing a coin occurrence of face '5' and occurrence of the head 'H' are independent events.

Multiplication law

If two events A and B are independent and P(A) and P(B) are their respective probability then the probability of their simultaneous occurrence is denoted by \( P(A \cap B) \) and defined as,
\(P(A\;and\;B) = P(A \cap B) = P(A) \times P(B). \)

Remarks:

• If A, B and C are three independent events then the probability of the simultaneous occurrence is given by \( P(A \cap B \cap C) = P(A) \times P(B) \times P(C) \)