Business StatisticsUnit 57 min read

Probability and Probability Distributions – Core Concepts

Unit 5 of Business Statistics: introduces probability fundamentals, event operations, conditional probability, Bayes theorem, discrete and continuous distributions, and their applications in business contexts.

Padhai · Based on the official syllabus and past papers
Updated 1 Oct 2026

Key points

Probability quantifies uncertainty; sample space and events form the foundation.
Conditional probability and independence determine how events influence each other.
Bayes theorem updates beliefs with new evidence.
Discrete distributions (binomial, Poisson, geometric, hypergeometric) model count data; continuous distributions (normal, uniform, exponential) model continuous outcomes.
Real‑world business decisions rely on probability models for risk assessment, forecasting, and quality control.

1. Probability Basics

Probability measures the likelihood of an event $E$ occurring within a sample space $Ω$ .
$P (E) = \frac{Number of outcomes in E}{Number of outcomes in Ω}$ where $0 \leq P (E) \leq 1$ .

1.1 Sample Space and Events

Symbol	Meaning	Example
$Ω$	Set of all possible outcomes	${1, 2, 3, 4, 5, 6}$ for a die
$E \subseteq Ω$	Event	Rolling an even number

1.2 Worked Example – Card Draw

A standard deck has 52 cards: 26 black, 26 red.

Probability of drawing a black card:
$P (black) = \frac{26}{52} = 0.5$
Probability of drawing a king:
$P (king) = \frac{4}{52} \approx 0.0769$

playing cards deck Standard deck of 52 playing cards (Image: Dbojan77, CC0, via Wikimedia Commons)

2. Operations on Events

Operation	Symbol	Formula	Example
Union	$A \cup B$	$P (A) + P (B) - P (A \cap B)$	Probability of drawing a heart or a king
Intersection	$A \cap B$	$P (A) P (B ∣ A)$	Probability of drawing a king that is also a heart
Complement	$A^{c}$	$1 - P (A)$	Probability of not drawing a spade

2.1 Venn Diagram

Intersection of Sets A and B: A ∩ B = {3}

3. Conditional Probability & Independence

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

Two events are independent if $P (A \cap B) = P (A) P (B)$ .

3.1 Worked Example – Scholarship

$P (boy gets scholarship) = 0.9$
$P (girl gets scholarship) = 0.8$

Assuming independence:
$P (at least one gets scholarship) = 1 - P (none) = 1 - (1 - 0.9) (1 - 0.8) = 1 - 0.02 = 0.98$

4. Bayes Theorem

Updates prior probabilities with new evidence.
$P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}$

4.1 Mermaid Flowchart

4.2 Real‑World Example – Credit Card Fraud

Prior probability of fraud $P (F) = 0.01$ .
Probability of a flagged transaction given fraud $P (T ∣ F) = 0.9$ .
Probability of a flagged transaction overall $P (T) = 0.05$ .

Posterior:
$P (F ∣ T) = \frac{0.9 \times 0.01}{0.05} = 0.18$

Thus, a flagged transaction has an 18 % chance of being fraudulent.

5. Discrete Probability Distributions

Distribution	Parameters	PMF	Typical Use
Binomial	$n, p$	$(k n) p^{k} (1 - p)^{n - k}$	Successes in $n$ trials
Poisson	$λ$	$\frac{e ^{- λ} λ ^{k}}{k !}$	Rare events per interval
Geometric	$p$	$(1 - p)^{k - 1} p$	First success time
Hypergeometric	$N, K, n$	$\frac{( k K ) ( n - k N - K )}{( n N )}$	Sampling without replacement

5.1 Worked Example – Daraz Order Queue

Assume Daraz receives on average $λ = 5$ orders per minute.
Probability of receiving exactly 3 orders in a minute:
$P (X = 3) = \frac{e ^{- 5} 5 ^{3}}{3 !} \approx 0.1404$

5.2 Binomial Example – Customer Satisfaction

A survey of 20 customers shows 15 are satisfied.
$n = 20, k = 15, p = 0.75$ .
$P (X = 15) = (15 20) (0.75)^{15} (0.25)^{5} \approx 0.044$

Binomial Distribution (n=5, p=0.8) for Customer Satisfaction

6. Continuous Probability Distributions

Distribution	PDF	Parameters	Typical Use
Uniform	$\frac{1}{b - a}$	$a, b$	Random time between events
Normal	$\frac{1}{σ 2 π} e^{- (x - μ)^{2} /2 σ^{2}}$	$μ, σ$	Measurement errors, salaries
Exponential	$λ e^{- λ x}$	$λ$	Time between arrivals

6.1 Worked Example – Delivery Time

Assume delivery times are normally distributed with $μ = 3$ days, $σ = 0.5$ .
Probability of delivery within 2.5 days:
$P (X \leq 2.5) = Φ (\frac{2.5 - 3}{0.5}) = Φ (- 1) = 0.1587$

Normal Distribution of Delivery Times

6.2 Exponential Example – Call Drop

Call drop rate $λ = 0.1$ per minute.
Probability that a call lasts more than 5 minutes:
$P (X > 5) = e^{- 0.1 \times 5} = e^{- 0.5} \approx 0.6065$

7. Applications in Business

Business Context	Probability Concept	How It Is Used
eSewa transaction success	Conditional probability	Estimate success rate given network conditions
Daraz order queue	Poisson distribution	Forecast peak order times
Ncell coverage	Binomial distribution	Calculate probability of coverage in a region
NEPSE price movement	Normal distribution	Value‑at‑risk calculations
Pathao delivery	Exponential distribution	Estimate average waiting time

7.1 Real‑World Example – eSewa Transaction Success

Let $S$ = successful transaction, $N$ = network stable.
$P (S ∣ N) = 0.99$ , $P (N) = 0.95$ .
Overall success probability:
$P (S) = P (S ∣ N) P (N) + P (S ∣ \overset{ˉ}{N}) P (\overset{ˉ}{N})$
Assuming $P (S ∣ \overset{ˉ}{N}) = 0.85$ :
$P (S) = 0.99 \times 0.95 + 0.85 \times 0.05 = 0.973$

Thus, eSewa can claim a 97.3 % success rate under typical conditions.

7.2 Real‑World Example – Ncell Coverage

A city has 1000 households. Survey shows 850 have coverage.
Probability that a randomly chosen household has coverage:
$P (C) = \frac{850}{1000} = 0.85$

If a new tower is planned, the company uses binomial distribution to estimate the number of households that will gain coverage.

8. Summary of Key Formulas

Concept	Formula
Probability of event	$P (E) = \frac{∣ E ∣}{∣Ω∣}$
Union	$P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Conditional	$P (B ∣ A) = \frac{P ( A \cap B )}{P ( A )}$
Independence	$P (A \cap B) = P (A) P (B)$
Bayes	$P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}$
Binomial	$(k n) p^{k} (1 - p)^{n - k}$
Poisson	$\frac{e ^{- λ} λ ^{k}}{k !}$
Normal	$\frac{1}{σ 2 π} e^{- (x - μ)^{2} /2 σ^{2}}$
Exponential	$λ e^{- λ x}$

In the Real World

Daraz Order Queue – Poisson distribution models the number of orders per minute, helping the logistics team allocate delivery staff during peak hours.
Ncell Network Coverage – Binomial distribution estimates the probability that a new tower will cover at least 90 % of households, guiding investment decisions.
NEPSE Price Movement – Normal distribution underpins value‑at‑risk calculations for investors, determining the probability of a 5 % drop in a day.

Exam Tip

Know the formulas: Write them on a cheat sheet; you’ll often be asked to apply them directly.
Work through worked examples: Practice with card, dice, and real‑world scenarios.
Draw trees and Venn diagrams: They help in conditional probability and event operations questions.
Use tables: For probability distributions, tabulate probabilities for small $k$ values to check your calculations.
Check assumptions: Independence, identical trials, and sample size are common traps.

Good luck!

Based on the TU BBS syllabus for Business Statistics (MGT207), unit 5.

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