Set Theory and Probability – Venn diagrams, counting, conditional probability, expectations

1. Sets and Set Operations

A set is a collection of distinct objects, called elements.
Notation: $A=\{a_1,a_2,\dots \}$; $x\in A$ means “$x$ is an element of $A$”.

1.1 Common Operations

1.2 Worked Example – Newspaper Survey

A survey of 500 students gave:

• $K$: read Kathmandu Post = 280
• $R$: read Rising Nepal = 190
• $H$: read Himalayan Times = 110
• $K\cap R=75$
• $R\cap H=50$
• $K\cap H=40$
• $K\cap R\cap H=20$

Find the number of students who read none of the three newspapers.

Solution (Inclusion–Exclusion):

$$\begin{array}{rl}|K\cup R\cup H|& =|K|+|R|+|H|\\ & -|K\cap R|-|K\cap H|-|R\cap H|\\ & +|K\cap R\cap H|\\ & =280+190+110-75-40-50+20=435.\end{array}$$

Students reading none:

$$\text{None}=500-435=65.$$

The diagram shows each region shaded with the exact count, making the inclusion–exclusion step visible.

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2. Counting Principles

2.1 Addition Principle

If a task can be performed in m ways or n ways (mutually exclusive), total ways = $m+n$.

2.2 Multiplication Principle

If a task consists of two independent stages, with m ways for stage 1 and n ways for stage 2, total ways = $m\times n$.

2.3 Permutations

Ordered arrangements of $r$ objects from $n$ distinct objects:

$$P(n,r)=\frac{n!}{(n-r)!}.$$

2.4 Combinations

Unordered selections of $r$ objects from $n$:

$$C(n,r)=\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}.$$

2.5 Worked Example – Daraz Order Queue

Daraz processes orders in three stages:

1. Pick a product (10 different items).
2. Choose a delivery slot (4 slots).
3. Select a payment method (3 options).

How many distinct order configurations are possible?

Using the multiplication principle:

$$10\times 4\times 3=120\text{ configurations}.$$

If the marketing team wants to know how many ways to choose 2 products for a bundle (order does not matter):

$$\left(\genfrac{}{}{0ex}{}{10}{2}\right)=45\text{ bundles}.$$

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3. Basics of Probability

3.1 Axioms

1. Non‑negativity: $P(A)\ge 0$.
2. Normalization: $P(U)=1$ where $U$ is the sample space.
3. Additivity: For mutually exclusive $A,B$, $P(A\cup B)=P(A)+P(B)$.

3.2 Classical Definition

If all outcomes are equally likely,

$$P(A)=\frac{|A|}{|U|}.$$

3.3 Worked Example – Lottery Ticket

A simple lottery draws 1 number from $\{1,2,\dots ,50\}$.

Probability of winning (choosing the exact number):

$$P(\text{win})=\frac{1}{50}=\mathrm{0.02.}$$

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4. Conditional Probability and Bayes Theorem

4.1 Conditional Probability

$$P(A|B)=\frac{P(A\cap B)}{P(B)},P(B)>0.$$

4.2 Multiplication Rule

$$P(A\cap B)=P(A|B)P(B)=P(B|A)P(A).$$

4.3 Bayes Theorem

$$P(A|B)=\frac{P(B|A)P(A)}{P(B)}.$$

4.4 Worked Example – Ncell Call Drop

Suppose Ncell reports:

• 5 % of all calls drop ($P(D)=0.05$).
• 30 % of calls are made during peak hours ($P(P)=0.30$).
• Among peak‑hour calls, 12 % drop ($P(D|P)=0.12$).

Find the probability that a dropped call occurred during peak hours, i.e., $P(P|D)$.

First compute $P(D\cap P)=P(D|P)P(P)=0.12\times 0.30=0.036$.

Then apply Bayes:

$$P(P|D)=\frac{P(D\cap P)}{P(D)}=\frac{0.036}{0.05}=\mathrm{0.72.}$$

So 72 % of dropped calls happen in peak hours.

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5. Discrete Random Variables

A random variable $X$ assigns a real number to each outcome of a random experiment.

5.1 Probability Mass Function (PMF)

For discrete $X$,

$$p_X(x)=P(X=x),\sum _x{p}_X(x)=1.$$

5.2 Expectation and Variance

$$E[X]=\sum _{x}x{p}_X(x),\mathrm{Var}(X)=\sum _x(x-E[X]{)}^2{p}_X(x).$$

5.3 Worked Example – Bank Loan Interest

A bank offers a fixed‑rate loan with interest rates (in % per annum) based on credit score:

Corresponding interest rates: 12, 10, 8, 6 respectively.

Find the expected interest rate.

$$\begin{array}{rl}E[R]& =12(0.25)+10(0.40)+8(0.20)+6(0.15)\\ & =3.0+4.0+1.6+0.9=9.5\%.\end{array}$$

Variance:

$$\begin{array}{rl}E[R^2]& =12^2(0.25)+10^2(0.40)+8^2(0.20)+6^2(0.15)\\ & =144(0.25)+100(0.40)+64(0.20)+36(0.15)\\ & =36+40+12.8+5.4=\mathrm{94.2.}\end{array}$$

$$\mathrm{Var}(R)=94.2-(9.5{)}^2=94.2-90.25=\mathrm{3.95.}$$

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6. Comparison of Counting Tools

| Tool                | Ordered? | Repetition allowed? | Formula                              | Typical use |
|---------------------|----------|---------------------|--------------------------------------|-------------|
| Permutations \(P(n,r)\) | Yes      | No                  | \(\frac{n!}{(n-r)!}\)                | Seating arrangements |
| Combinations \(C(n,r)\) | No       | No                  | \(\binom{n}{r}\)                     | Selecting a committee |
| Variations with repetition | Yes      | Yes                 | \(n^{r}\)                            | Password generation |
| Multiset combinations | No       | Yes                 | \(\binom{n+r-1}{r}\)                 | Distributing identical items |

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7. Applications in the Real World

7.1 eSewa – Set Operations

When a user selects multiple payment methods (eSewa wallet, credit card, bank transfer), the system internally treats the set of available methods as $M=\{\text{Wallet},\text{Card},\text{Bank}\}$.

• Union of promotions: $P_{\text{discount}}=P_{\text{festival}}\cup P_{\text{new‑user}}$.
• Intersection for eligibility: a user must belong to both registered and verified sets to use a high‑value transfer.

7.2 Daraz – Counting & Probability

Daraz’s flash sale offers 5 limited‑stock items. The probability that a randomly arriving customer gets the first item is

$$P(\text{first item})=\frac{1}{5}=\mathrm{0.20.}$$

If the sale runs for 3 hours and each hour 200 customers arrive (independent), the expected number of first‑item sales is

$$E=200\times 3\times 0.20=120.$$

7.3 Ncell – Conditional Probability

Network engineers use conditional probability to predict call drop during peak hours (as shown in the worked example). The result guides capacity planning: if $P(P|D)=0.72$, then upgrading peak‑hour towers reduces overall drop rate most efficiently.

7.4 NEPSE – Expectation

Investors model daily price changes $X$ as a discrete random variable with probabilities derived from historical frequencies. The expected daily return $E[X]$ informs portfolio decisions.

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8. Summary of Key Formulas

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In the real world

1. eSewa uses set theory to manage eligible payment options: a user belongs to the set Verified ∩ Has Wallet Balance to use the instant transfer feature.
2. Daraz applies counting principles when generating bundle offers: choosing 3 items from 12 best‑sellers gives $\left(\genfrac{}{}{0ex}{}{12}{3}\right)=220$ possible bundles, each displayed as a separate product page.
3. Ncell relies on conditional probability to allocate extra bandwidth: if $P(\text{drop}|\text{peak})=0.12$ and peak traffic occupies 30 % of total time, the network team predicts that 72 % of all drops happen during peak, prompting targeted upgrades.

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Exam tip

• Venn diagrams are worth 2 marks each; always label every region with the exact count before applying inclusion–exclusion.
• For probability questions, write the sample space size first, then apply the classical formula; this prevents arithmetic slips.
• When a problem mentions “ordered” vs “unordered”, instantly decide between permutations and combinations—the exam often tests this distinction.
• For conditional probability, sketch a two‑branch tree (use the provided tree figure as a template) to keep track of $P(A\cap B)$ and $P(B)$.
• Remember the expectation shortcut: $E[X]=\sum xp(x)$. If the PMF is symmetric (e.g., fair dice), you can use the known mean (3.5 for a die) to save time.

Focus on clear notation, step‑by‑step calculations, and a quick visual check (Venn, tree, or bar chart) before writing the final answer. Good luck!