Probability Theory: Events, Rules, Distributions & Applications

TAKEAWAYS

• Probability is the mathematical measure of uncertainty (0 to 1) for events, calculated via classical, empirical, or subjective methods.
• Key rules: Addition (mutually exclusive events), Multiplication (independent events), and Complementary probability ($P(A)=1-P(A^{\prime })$).
• Conditional probability ($P(A|B)=\frac{P(A\cap B)}{P(B)}$) and Bayes’ Theorem reverse probabilities for real-world decisions (e.g., fraud detection in Khalti).
• Discrete distributions (Binomial, Poisson) model countable outcomes (e.g., number of defective Daraz orders), while continuous distributions (Normal) model measurements (e.g., NTC call durations).
• Expected value ($E(X)=\sum x\cdot P(x)$) and variance (${\sigma }^2=E(X^2)-[E(X){]}^2$) quantify risk in business (e.g., loan defaults in NMB Bank).
• Visual tools: Venn diagrams for events, probability trees for sequential events, and normal distribution curves for real-world data (e.g., NEPSE stock returns).

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1. Foundations of Probability

1.1 Definitions and Sample Space

Probability quantifies the likelihood of an event occurring. The sample space (S) is the set of all possible outcomes of an experiment.

Example 1.1: Tossing a coin.

• Sample space (S): $\{H,T\}$
• Event (E): Getting a head.
• Probability of E: $P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}=\frac{1}{2}$

1.2 Types of Probability

Worked Example 1.2: A bag has 3 red, 2 black, and 5 white balls. What is the probability of drawing:

1. A red ball?
2. A non-white ball?

Solution:

• Total balls = $3+2+5=10$.
• $P(\text{Red})=\frac{3}{10}=0.3$.
• $P(\text{Non-white})=P(\text{Red or Black})=\frac{3+2}{10}=0.5$.

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2. Probability Rules and Theorems

2.1 Addition Rule

For any two events A and B: $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

• If mutually exclusive ($A\cap B=\varnothing$), then $P(A\cup B)=P(A)+P(B)$.

Example 2.1: Probability of drawing a multiple of 3 or 7 from balls 1–20.

• Multiples of 3: $\{3,6,9,12,15,18\}$ → 6 balls.
• Multiples of 7: $\{7,14\}$ → 2 balls.
• Intersection: $\{14\}$ (only 1 ball).
• $P(3\text{ or }7)=\frac{6}{20}+\frac{2}{20}-\frac{1}{20}=\frac{7}{20}$.

2.2 Multiplication Rule

For independent events (e.g., coin tosses, card draws without replacement): $$P(A\cap B)=P(A)\times P(B)$$ For dependent events: $$P(A\cap B)=P(A)\times P(B|A)$$

Example 2.2: Probability of two red balls drawn without replacement from the same bag (3 red, 2 black, 5 white).

• $P(\text{First red})=\frac{3}{10}$.
• $P(\text{Second red}|\text{First red})=\frac{2}{9}$.
• $P(\text{Both red})=\frac{3}{10}\times \frac{2}{9}=\frac{6}{90}=\frac{1}{15}$.

2.3 Conditional Probability and Bayes’ Theorem

Conditional Probability: $$P(A|B)=\frac{P(A\cap B)}{P(B)}$$ Bayes’ Theorem: $$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$$

Real-World Example 2.3: eSewa Fraud Detection

• Suppose 1% of eSewa transactions are fraudulent ($P(F)=0.01$).
• 90% of fraudulent transactions have high transaction value ($P(H|F)=0.9$).
• 5% of all transactions are high-value ($P(H)=0.05$).
• What is the probability a high-value transaction is fraudulent? P(F|H) = \frac{P(H|F) \cdot P(F)}{P(H)} = \frac{0.9 \times 0.01}{0.05} = 0.18 \text{ (18%)}

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3. Probability Distributions

3.1 Discrete Distributions

Binomial Distribution

Models exactly $k$ successes in $n$ independent trials (e.g., defective products, survey responses). $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$

• Mean: $\mu =np$
• Variance: ${\sigma }^2=np(1-p)$

Example 3.1: Daraz Order Defects

• Probability of a defective order: $p=0.05$.
• Find the probability that exactly 2 out of 10 orders are defective. P(X = 2) = \binom{10}{2} (0.05)^2 (0.95)^8 = 45 \times 0.0025 \times 0.6634 \approx 0.0746 \text{ (7.46%)}

Poisson Distribution

Models rare events over a fixed interval (e.g., customer complaints, machine failures). $$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$$

• Mean: $\mu =\lambda$
• Variance: ${\sigma }^2=\lambda$

Example 3.2: Ncell Customer Complaints

• Average complaints per hour: $\lambda =3$.
• Probability of exactly 5 complaints in an hour: P(X = 5) = \frac{e^{-3} 3^5}{5!} = \frac{243}{120} \times 0.0498 \approx 0.1008 \text{ (10.08%)}

3.2 Continuous Distributions

Normal Distribution

Models continuous data (e.g., heights, incomes, NEPSE stock prices).

• PDF: $f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$
• 68-95-99.7 Rule:
  • 68% within $\mu \pm \sigma$
  • 95% within $\mu \pm 2\sigma$
  • 99.7% within $\mu \pm 3\sigma$

Example 3.3: NEPSE Stock Returns

• Mean return: $\mu =10\%$, Standard deviation: $\sigma =4\%$.
• Probability of a return between 6% and 14%:
  • Convert to Z-scores: $$Z_1=\frac{6-10}{4}=-1,Z_2=\frac{14-10}{4}=1$$
  • From Z-table: $P(-1<Z<1)=0.6826$ (68.26%).

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## In the Real World

1. eSewa Transaction Risk

   • Idea Used: Conditional Probability & Bayes’ Theorem
   • How: eSewa uses historical data to calculate $P(\text{Fraud}|\text{High Value})$ to flag suspicious transactions. For example, if 1% of transactions are fraudulent but 90% of frauds involve amounts > Rs 50,000, Bayes’ Theorem helps prioritize investigations.

2. Daraz Order Fulfillment

   • Idea Used: Binomial Distribution
   • How: Daraz estimates the probability of exactly $k$ defective orders in a batch of 100 using the binomial formula. If $p=0.02$ (2% defect rate), they calculate $P(X=3)$ to plan quality checks.

3. Ncell Network Traffic

   • Idea Used: Poisson Distribution
   • How: Ncell models call arrivals per minute as a Poisson process ($\lambda =15$ calls/minute). They compute $P(X>20)$ to upgrade servers during peak hours (e.g., 6–9 PM).

4. NMB Bank Loan Approvals

   • Idea Used: Normal Distribution
   • How: Loan default rates are assumed normal ($\mu =5\%$, $\sigma =1.5\%$). The bank calculates $P(X>8\%)$ to adjust interest rates for high-risk borrowers.

5. Khalti Merchant Payouts

   • Idea Used: Expected Value
   • How: Khalti computes the average payout delay ($E(X)$) for merchants. If delays are $X$ days with $P(X=1)=0.6$, $P(X=2)=0.3$, $P(X=3)=0.1$, then: $$E(X)=1\times 0.6+2\times 0.3+3\times 0.1=1.5\text{ days}$$ This helps set merchant expectations.

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4. Expected Value and Variance

4.1 Expected Value

The long-term average of a random variable. $$E(X)=\sum x\cdot P(x)\text{(Discrete)}\text{or}E(X)=\int xf(x)dx\text{(Continuous)}$$

Example 4.1: NEPSE Stock Investment

$$E(X)=5\times 0.2+10\times 0.5+15\times 0.3=1+5+4.5=10.5\%$$

4.2 Variance and Standard Deviation

Measures spread of data. $$\text{Variance}={\sigma }^2=E(X^2)-[E(X){]}^2$$ $$\text{Standard Deviation}=\sigma =\sqrt{{\sigma }^2}$$

Example 4.2: Bank Employee Salaries

1. Calculate $E(X)$: $$E(X)=40,000\times 0.3+50,000\times 0.5+60,000\times 0.2=49,000$$
2. Calculate $E(X^2)$: $$E(X^2)=40,000^2\times 0.3+50,000^2\times 0.5+60,000^2\times 0.2=2,410,000,000$$
3. Variance: $${\sigma }^2=2,410,000,000-(49,000{)}^2=1,501,000(\sigma \approx 1,225)$$

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## Exam Tip

1. Memorize Key Formulas:

   • Addition Rule, Multiplication Rule, Bayes’ Theorem, Binomial/Poisson/Normal formulas.
   • Shortcut: For normal distribution, always convert to Z-scores before using tables.

2. Identify Problem Types:

   • With/Without Replacement: Affects probability calculations (e.g., drawing balls vs. tossing coins).
   • Independent/Dependent Events: Use $P(A\cap B)=P(A)P(B)$ only for independent events.

3. Visualize Probabilities:

   • Draw Venn diagrams for events, trees for sequential probabilities, and normal curves for continuous data.
   • Example: For a question on "probability of A or B," sketch a Venn diagram to avoid missing intersections.

4. Real-World Applications:

   • Business Context: Always relate to risk assessment (e.g., loan defaults), quality control (defective products), or decision-making (e.g., eSewa fraud).
   • Nepal Focus: Use examples from NEPSE, Ncell, Daraz, or banks to score extra marks for relevance.

5. Common Pitfalls:

   • Ignoring "without replacement": Probabilities change after each draw (e.g., second ball draw).
   • Misapplying Bayes’ Theorem: Ensure you correctly identify $P(A|B)$ vs. $P(B|A)$.
   • Normal Approximation: Only use for large $n$ in binomial distributions ($np\ge 5$ and $n(1-p)\ge 5$).

6. Worked Example Strategy:

   • Step 1: Define the sample space and events clearly.
   • Step 2: Identify whether events are independent/dependent.
   • Step 3: Apply the correct rule (addition/multiplication).
   • Step 4: For distributions, state the parameters ($\mu ,\sigma ,n,p$) before calculations.

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In the real world

• eSewa Fraud Detection: Uses Bayes’ Theorem to calculate the probability of fraud given high-value transactions (P(F|H) = 18%), helping the company flag suspicious activities and reduce financial losses.
• Daraz Quality Control: Applies the Binomial distribution to estimate the likelihood of defective orders (e.g., P(X=2) = 7.46% for 2 defective orders out of 10), ensuring better customer satisfaction and reducing returns.
• NEPSE Stock Analysis: Relies on the Normal distribution to model stock returns (μ=10%, σ=4%), helping investors assess risks and make informed decisions about buying or selling shares.