Probability Models: Distributions, Rules & Real-World Applications

Key Concepts and Definitions

1. Probability Basics

Probability quantifies the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain). It is the foundation for all probability models.

Probability Rules

• Addition Rule: For two events $A$ and $B$, $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$ If $A$ and $B$ are mutually exclusive (cannot occur simultaneously), $P(A\cap B)=0$, so: $$P(A\cup B)=P(A)+P(B)$$

• Multiplication Rule: For two events $A$ and $B$, $$P(A\cap B)=P(A)\cdot P(B|A)$$ If $A$ and $B$ are independent, $P(B|A)=P(B)$, so: $$P(A\cap B)=P(A)\cdot P(B)$$

• Complement Rule: $$P(A^c)=1-P(A)$$

2. Discrete vs. Continuous Probability Distributions

Probability distributions describe how probabilities are assigned to possible outcomes.

Discrete Distributions

Used for countable outcomes (e.g., number of customers, defects).

• Binomial Distribution: Models the number of successes ($k$) in $n$ independent trials, each with success probability $p$. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$ Example: Probability that exactly 3 out of 10 customers use Khalti for a transaction, given a 20% success rate. $$P(X=3)=\left(\genfrac{}{}{0ex}{}{10}{3}\right)(0.2{)}^3(0.8{)}^7=28.56\%$$

• Poisson Distribution: Models rare events over a fixed interval (e.g., call drops per hour). $$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$$ Example: Ncell records an average of 5 call drops per hour. What is the probability of 3 drops in an hour? $$P(X=3)=\frac{e^{-5}{5}^3}{3!}=17.55\%$$

Continuous Distributions

Used for uncountable outcomes (e.g., height, time).

• Normal Distribution: Bell-shaped, symmetric around the mean ($\mu$), with standard deviation ($\sigma$). $$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$$ Example: NTC measures internet speeds in Kathmandu with $\mu =20$ Mbps and $\sigma =5$ Mbps. What is the probability a randomly selected user gets >25 Mbps? Use the Z-score: $$Z=\frac{25-20}{5}=1$$ From the standard normal table, $P(Z>1)=16.0\%$.

• Uniform Distribution: All outcomes equally likely (e.g., random selection of a customer from a list). $$f(x)=\frac{1}{b-a}\text{for}a\le x\le b$$

In the Real World

1. Daraz’s Inventory Management Daraz uses the Poisson distribution to predict how many units of a product (e.g., a smartphone) will be sold in a day. If the average daily sales are 50 units ($\lambda =50$), Daraz can stock enough inventory to meet demand with 95% confidence by calculating: $$P(X\le 70)=95\%\text{(using Poisson tables)}$$ This reduces stockouts and overstocking.

2. Ncell’s Call-Drop Prediction Ncell analyzes call drops using the Binomial distribution to identify high-risk areas. If a tower has a 10% call-drop rate and 100 calls are logged, the probability of >15 drops is: $$P(X>15)=1-P(X\le 15)=1-0.987=1.3\%$$ This helps prioritize maintenance.

3. Khalti’s Fraud Detection Khalti uses Bayes’ Theorem to update fraud probabilities. If 5% of transactions are fraudulent and a new detection model flags a transaction with 80% accuracy, the posterior probability of fraud given a flag is: $$P(\text{Fraud}|\text{Flag})=\frac{P(\text{Flag}|\text{Fraud})\cdot P(\text{Fraud})}{P(\text{Flag})}=\frac{0.8\times 0.05}{0.8\times 0.05+0.2\times 0.95}=21.05\%$$ This balances false positives and negatives.

3. Bayes’ Theorem

Updates probabilities based on new evidence: $$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$$ Example: Pathao wants to predict rider cancellations. Historically, 10% of riders cancel. If a rider books late (after 8 PM), the cancellation rate jumps to 30%. What is the probability a rider cancels given they booked late? $$P(\text{Cancel}|\text{Late})=\frac{P(\text{Late}|\text{Cancel})\cdot P(\text{Cancel})}{P(\text{Late})}=\frac{0.8\times 0.1}{0.3}=26.67\%$$ (Assumes 80% of cancellations occur late and 30% of late bookings cancel.)

4. Probability Trees and Decision Analysis

Visualize sequential probabilities using probability trees.

graph TD
    A["Start"] --> B["Event 1: Success (p=0.6)"]
    A --> C["Event 1: Failure (p=0.4)"]
    B --> D["Event 2: Success (p=0.7)"]
    B --> E["Event 2: Failure (p=0.3)"]
    C --> F["Event 2: Success (p=0.5)"]
    C --> G["Event 2: Failure (p=0.5)"]
    D --> H["Outcome: Both Success (0.6*0.7=0.42)"]
    E --> I["Outcome: Event 1 Success, Event 2 Failure (0.6*0.3=0.18)"]
    F --> J["Outcome: Event 1 Failure, Event 2 Success (0.4*0.5=0.20)"]
    G --> K["Outcome: Both Failure (0.4*0.5=0.20)"]

Example: NEPSE tracks stock price movements. If a stock has a 60% chance of rising tomorrow and a 40% chance of falling, and rising stocks have a 70% chance of rising further the next day, what is the probability the stock rises two days in a row? P(\text{Rise Day 1 and Day 2}) = 0.6 \times 0.7 = 0.42 \quad \text{(42%)}

5. Joint and Marginal Probabilities

• Joint Probability: $P(A\cap B)$ = probability both $A$ and $B$ occur.
• Marginal Probability: $P(A)$ = probability of $A$ regardless of $B$.

Example: Bank Loan Approvals A bank approves 60% of loans, and 30% of approved loans default. What is the probability a randomly selected loan is approved and defaults? P(\text{Approved} \cap \text{Default}) = P(\text{Default}|\text{Approved}) \cdot P(\text{Approved}) = 0.3 \times 0.6 = 0.18 \quad \text{(18%)}

Comparison Table: Key Probability Distributions

Advantages and Limitations

Worked Example: Traffic Congestion Prediction (Kathmandu)

Scenario: The NTC monitors traffic on Ring Road. Historical data shows:

• 60% chance of congestion during peak hours (7–9 AM).
• If congested, 70% chance of delays >30 minutes.
• If not congested, 20% chance of delays >30 minutes.

Question: What is the probability a commuter faces >30-minute delays during peak hours?

Solution:

1. Define Events:

   • $C$ = Congestion occurs ($P(C)=0.6$)
   • $D$ = Delay >30 minutes
   • $P(D|C)=0.7$, $P(D|C^c)=0.2$

2. Use Law of Total Probability: $$P(D)=P(D|C)\cdot P(C)+P(D|C^c)\cdot P(C^c)$$ P(D) = (0.7 \times 0.6) + (0.2 \times 0.4) = 0.42 + 0.08 = 0.50 \quad \text{(50%)}

Interpretation: There is a 50% chance of >30-minute delays during peak hours on Ring Road.

Exam Tip

1. Memorize Key Formulas:

   • Binomial: $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$
   • Poisson: $P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$
   • Bayes’ Theorem: $P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$
   • Normal: Use Z-scores and standard normal tables.

2. Practice Probability Trees:

   • Draw trees for sequential events (e.g., Pathao’s rider cancellations).
   • Label branches with probabilities and outcomes.

3. Real-World Applications:

   • Business: Use Binomial/Poisson for risk assessment (e.g., Daraz’s inventory).
   • Finance: Apply Normal distribution to stock returns (e.g., NEPSE).
   • Operations: Use Bayes’ Theorem for fraud detection (e.g., Khalti).

4. Common Pitfalls:

   • Independence vs. Dependence: Assume independence only if stated.
   • Discrete vs. Continuous: Use the correct distribution (e.g., Poisson for counts, Normal for measurements).
   • Complement Rule: Always check if $P(A^c)$ is easier to calculate.

5. Exam Strategy:

   • Part A (Short Answers): Define terms like joint probability, marginal probability, and Bayes’ Theorem.
   • Part B (Calculations): Show all steps for binomial/Poisson/normal problems.
   • Part C (Applications): Relate to Nepali businesses (e.g., Ncell’s call drops, Daraz’s sales forecasting).

Visual Summary: Probability Distributions in Business

mindmap
  root((Probability Models in Business))
    Binomial["Binomial Distribution<br>• Success/Failure<br>• Example: Khalti transactions"]
    Poisson["Poisson Distribution<br>• Rare Events<br>• Example: Ncell call drops"]
    Normal["Normal Distribution<br>• Symmetric Data<br>• Example: NTC internet speeds"]
    Bayes["Bayes’ Theorem<br>• Updating Probabilities<br>• Example: Khalti fraud detection"]
    Trees["Probability Trees<br>• Sequential Events<br>• Example: Pathao cancellations"]

Real-World Image: Khalti Transaction Flow