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Unit 4 of 4 · BBA Sem 2

Unit 4: Probability & probability distributions

Business Statistics notes · PTU syllabus (BBA 201-18)

3 min read4 topics10 exam questions
On this page
  1. Unit summary
  2. Approaches to probability
  3. Laws of probability
  4. Bayes' theorem
  5. Binomial, Poisson and normal distributions
  6. Key terms
  7. Quick revision
  8. Important questions

Unit summary

Business is full of uncertainty — will a customer buy, will a machine fail? Probability measures uncertainty, and probability distributions model it. This unit covers the approaches to probability, addition and multiplication laws, conditional probability, Bayes' theorem, and the binomial, Poisson and normal distributions.

After this unit you can

  • Explain the classical, relative frequency and subjective approaches to probability
  • Apply the addition and multiplication laws and conditional probability
  • Apply Bayes' theorem
  • Solve problems using the binomial, Poisson and normal distributions

PTU syllabus topics

  • Approaches to calculating probability
  • addition and multiplication laws
  • conditional probability and Bayes' Theorem
  • Binomial
  • Poisson and Normal distributions
Key formulasProbability rules and distributions

  • Addition law

    P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

  • Multiplication law

    P(A ∩ B) = P(A) × P(B given A)

  • Bayes' theorem

    P(A given B) = P(B given A) P(A) / P(B)

  • Binomial

    P(r) = nCr p^r q^(n−r)

  • Poisson

    P(r) = e^(−λ) λ^r / r!

1

Topic 1

Approaches to probability

Probability is a number between 0 and 1 measuring the chance that an event will occur.

  • Classical: P(A) = favourable outcomes / total equally likely outcomes.
  • Relative frequency (empirical): P(A) = number of times A occurred / number of trials.
  • Subjective: based on personal judgement and experience.
  • Axiomatic: based on axioms (Kolmogorov) — probability is non-negative, P(sample space) = 1, and additive for mutually exclusive events.
2

Topic 2

Laws of probability

Key formulasLaws of probability

  • Addition law

    P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

  • Mutually exclusive events

    P(A ∪ B) = P(A) + P(B)

  • Multiplication law

    P(A ∩ B) = P(A) × P(B given A)

  • Independent events

    P(A ∩ B) = P(A) × P(B)

  • Conditional probability

    P(A given B) = P(A ∩ B) / P(B)

Example

A card is drawn from a pack of 52. P(king or heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.

3

Topic 3

Bayes' theorem

Bayes' theorem revises probabilities when new information arrives: P(Aᵢ given B) = P(Aᵢ) P(B given Aᵢ) / Σ P(Aⱼ) P(B given Aⱼ)

Example

Machines A and B make 60% and 40% of output, with defect rates 2% and 5%. A defective item is found. P(from A) = (0.6 × 0.02) / (0.6 × 0.02 + 0.4 × 0.05) = 0.012/0.032 = 0.375.

4

Topic 4

Binomial, Poisson and normal distributions

ComparisonProbability distributions
Binomial
Poisson

Use

Fixed n trials, two outcomes, constant p

Rare events in an interval

Formula

P(r) = nCr pʳ qⁿ⁻ʳ

P(r) = e⁻ᵐ mʳ / r!

Mean

np

m

Variance

npq

m

Normal distribution: a continuous, bell-shaped, symmetric distribution with mean = median = mode. Probabilities are found with Z = (X − μ)/σ and the normal table. About 68.27% of values lie within μ ± 1σ, 95.45% within μ ± 2σ and 99.73% within μ ± 3σ.

Example

Marks are normal with μ = 60 and σ = 10. For X = 75, Z = 1.5; the area from 0 to 1.5 is 0.4332, so P(X > 75) = 0.5 − 0.4332 = 0.0668 (about 6.7%).

Key terms

Probability
A measure of the likelihood of an event, from 0 to 1
Mutually exclusive events
Events that cannot occur together
Conditional probability
Probability of A given that B has occurred
Bayes' theorem
Revising probabilities using new evidence
Normal distribution
A symmetric bell-shaped continuous distribution

Quick revision

  • Addition: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
  • Multiplication: P(A ∩ B) = P(A)P(B given A).
  • Binomial mean np, variance npq; Poisson mean = variance = m.
  • Z = (X − μ)/σ; 68–95–99.7 rule.

Important exam questions

Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).

Short-answer questions

  1. Q1.Define probability and its classical approach.
  2. Q2.State the addition law of probability.
  3. Q3.What are independent events?
  4. Q4.State Bayes' theorem.
  5. Q5.Write the mean and variance of a binomial distribution.
  6. Q6.State two properties of the normal distribution.

Long-answer questions

  1. Q1.Explain the approaches to probability and the addition and multiplication laws with examples.
  2. Q2.Solve a problem using Bayes' theorem.
  3. Q3.Explain the binomial and Poisson distributions and solve one problem on each.
  4. Q4.Explain the normal distribution and its properties, and solve a problem using the Z table.

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