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Probability Handwritten Notes PDF
Chapter 13: Statistics Notes
Key Points Probability
Probability
• Random experiment – An experiment whose outcomes cannot be predicted or determined in advance is called a random experiment.
• Outcome – A possible result of a random experiment is called its outcome.
• Sample space – The set of all possible outcomes.
• Sample points – Elements of sample space.
• Events – An event is a subset of a sample space associated with a random experiment.
Types of events
• Impossible events – The empty set.
• Sure events – The whole sample space.
• Complementary event or not event – The set A′ or S − A.
Event A or B : The set A ∪ B
Event A and B : The set A ∩ B
Event A and not B : The set A − B
• Mutually exclusive event
A and B events are mutually exclusive if both events cannot occur simultaneously, i.e.
A ∩ B = φ or
P(A ∩ B) = 0
• Exhaustive events
If E₁, E₂, … En are n events of a sample space S and if
E₁ ∪ E₂ ∪ E₃ ∪ … ∪ En = S
then E₁, E₂, … En are called exhaustive events.
• Exhaustive and mutually exclusive events
Events E₁, E₂, … En are mutually exclusive and exhaustive if
E₁ ∪ E₂ ∪ … ∪ En = S
and
Eᵢ ∩ Eⱼ = φ ∀ i ≠ j
• Probability
Let S = {ω₁, ω₂, … ωn} be the sample space associated with a random experiment. Then, a function P which assigns every event A ⊆ S to a unique non-negative real number P(A) is called the probability function.
Number P(ωᵢ) associated with sample point ωᵢ such that:
(i) 0 ≤ P(ωᵢ) ≤ 1 for each ωᵢ ∈ S
(ii) Σ P(ωᵢ) = 1 (for all ωᵢ ∈ S)
(iii) P(A) = Σ P(ωᵢ) for all ωᵢ ∈ A
The number P(ωᵢ) is called probability of the outcome ωᵢ.