Basics of Probability and Sampling Distributions: Normal Distribution Overview and Applications

Overview

Probability is the foundation of business statistics. It helps measure uncertainty in demand, quality, finance, risk and decision-making. This topic covers basic probability laws, conditional probability, Bayes’ theorem, and the normal distribution. It also introduces sampling distributions and CLT, which are used in estimation and hypothesis testing.

Probability: meaning and notation

Probability is a numerical measure of the likelihood of an event. For an event $A$, probability is written as $P(A)$, and it always satisfies $0 \\le P(A) \\le 1$.

Basic terms (experiment, outcome, event, sample space)

• Random experiment: process with uncertain outcome (toss a coin, inspect an item).
• Outcome: result of an experiment (Head/Tail).
• Sample space (S): set of all possible outcomes.
• Event (A): subset of S (e.g., “Head”).

Types of probability (brief)

• Classical probability: based on equally likely outcomes.
• Empirical probability: based on observed frequency (past data).
• Subjective probability: based on judgment/experience.

Basic laws of probability

• $0 \\le P(A) \\le 1$
• $P(S) = 1$, $P(\\varnothing)=0$
• Complement: $P(A') = 1 - P(A)$

Addition rule and multiplication rule

Addition rule: $P(A \\cup B) = P(A) + P(B) - P(A \\cap B)$ If A and B are mutually exclusive, then $P(A \\cap B)=0$ so $P(A\\cup B)=P(A)+P(B)$.

Multiplication rule: $P(A \\cap B) = P(A)\\,P(B\\mid A)$ If A and B are independent, then $P(B\\mid A)=P(B)$ so $P(A\\cap B)=P(A)P(B)$.

Conditional probability

Conditional probability of A given B: $P(A\\mid B) = \\frac{P(A \\cap B)}{P(B)},\\quad P(B)>0$

Bayes’ theorem (idea and use)

Bayes’ theorem helps update probabilities when new information is available: $P(A\\mid B) = \\frac{P(B\\mid A)P(A)}{P(B)}$ In business, Bayes is used in quality control (defect detection), credit risk, and decision-making under uncertainty.

Random variable and probability distribution (concept)

A random variable (X) assigns a numerical value to each outcome.
A probability distribution shows probabilities of possible values of X.

Normal distribution: features

Normal distribution is a continuous, bell-shaped distribution with:

• symmetric about mean $\\mu$
• mean = median = mode
• defined by parameters $\\mu$ and standard deviation $\\sigma$
• total area under curve = 1

Empirical rule (approx.):

• about 68% within $\\mu \\pm 1\\sigma$
• about 95% within $\\mu \\pm 2\\sigma$
• about 99.7% within $\\mu \\pm 3\\sigma$

Standard normal (Z) and z-score

To convert a normal variable X to standard normal Z: $Z = \\frac{X-\\mu}{\\sigma}$ Z-tables give area/probability for Z values.

Sampling distributions (mean and proportion)

Sampling distribution is the probability distribution of a sample statistic (like sample mean $\\bar X$) over repeated samples.

Key results (concept level):

• Mean of sample means: $E(\\bar X)=\\mu$
• Standard error of mean: $SE(\\bar X)=\\sigma/\\sqrt{n}$ (when population $\\sigma$ known)

For sample proportion $\u0302p$:

• $E(\\hat p)=p$
• $SE(\\hat p)=\\sqrt{\\frac{p(1-p)}{n}}$

Central Limit Theorem (CLT): overview

CLT states that for large n, the sampling distribution of the sample mean tends to be approximately normal (even if the population is not perfectly normal), with mean $\\mu$ and standard error $\\sigma/\\sqrt{n}$. This is the basis for many statistical methods used in business.

Applications in business

• quality control and defect probability
• demand forecasting under uncertainty
• risk analysis and insurance
• inventory decisions and service levels
• sampling inspection and market research

Formula table (quick)

Problem approach flowchart (exam)

Common exam mistakes

• using addition rule without subtracting intersection term
• confusing mutually exclusive with independent events
• applying Bayes without writing prior and conditional probabilities clearly
• wrong z-score sign (X-μ) and wrong σ vs SE

Key Terms

• Event: subset of sample space.
• Independent events: occurrence of one does not affect the other.
• Mutually exclusive: cannot occur together.
• Conditional probability: probability given another event occurred.
• Normal distribution: bell-shaped continuous distribution.
• Sampling distribution: distribution of a statistic across samples.

Quick Recap (1-minute revision)

• Probability laws: complement, addition, multiplication, conditional and Bayes.
• Normal distribution: symmetric, defined by μ and σ; use z-score.
• Sampling distributions + CLT: basis for estimation and testing.