Important Formulas Data Handling - (Maths) Class 7 (Old NCERT)

Important Formulas

(1) Trial - A trial is an action which results in one or several outcomes.

(2) Random experiment - An experiment in which the result of a trial cannot be predicted in advance is called a random experiment.

(3) Event - An event associated with a random experiment is the collection of some outcomes of the experiment.

(4) An event associated with a random experiment is said to happen (or occur) if any one of the outcomes that form the event is obtained when the experiment is performed.

(5) The empirical probability (also called experimental or relative-frequency probability) of happening of an event E is defined as the ratio of the number of times the event E occurs to the total number of trials performed.

Key terms and notation

• Outcome - A single possible result of a trial.
• Sample space (denoted by S) - The set of all possible outcomes of a random experiment.
• Favourable outcomes - Outcomes that satisfy the definition of the event whose probability is sought.
• Event - A subset of the sample space; may contain one or more outcomes.
• P(E) - Probability of event E happening.

Formulas

• Empirical probability
  P(E) ≈ (Number of times E occurred) ÷ (Total number of trials)
  This is based on observed frequencies when an experiment is repeated many times.
• Theoretical probability (for equally likely outcomes)
  P(E) = (Number of favourable outcomes) ÷ (Total number of possible outcomes in S)
  This is used when all outcomes in the sample space are equally likely, for example a fair coin or a fair die.
• Complement rule
  If A' denotes the complement of event A (A does not happen), then
  P(A') = 1 − P(A)
• Bounds on probability
  For any event E,
  0 ≤ P(E) ≤ 1
• Sure and impossible events
  If S is the sample space then P(S) = 1.
  If ∅ is the empty set (no outcome) then P(∅) = 0.

Types of events 

• Elementary event - An event consisting of exactly one outcome.
• Compound event - An event consisting of more than one outcome.
• Complementary events - Two events are complementary if one happens exactly when the other does not (A and A').
• Mutually exclusive events - Events that cannot happen at the same time (their intersection is empty).

Relation between empirical and theoretical probability

• When the number of trials is very large, the empirical probability of an event tends to approach its theoretical probability, provided the trials are performed under similar conditions.
• Empirical probability is useful when theoretical calculation is difficult or when the experiment truly reflects real-world randomness.