Probability: Rules, Events, Distributions & Real Applications

TAKEAWAYS

• Probability quantifies uncertainty using values between 0 (impossible) and 1 (certain), calculated as $P(E)=\frac{\text{favorable outcomes}}{\text{total possible outcomes}}$.
• Key rules: Addition (mutually exclusive events), Multiplication (independent events), and Complement (1 – $P(E)$) govern how probabilities combine.
• Conditional probability ($P(A|B)$) and Bayes’ Theorem ($P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$) solve real-world problems like fraud detection (e.g., Khalti flagging suspicious transactions).
• Probability distributions (discrete vs. continuous) model scenarios: Binomial for fixed trials (e.g., Ncell customer churn), Poisson for rare events (e.g., Daraz delivery delays), and Normal for symmetric data (e.g., NEPSE stock returns).
• Expected value ($E(X)=\sum x\cdot P(x)$) helps businesses decide on investments (e.g., banks calculating loan default risks).
• Exam focus: Solve problems using Venn diagrams, probability trees, and formulas—never guess. Always check if events are independent or mutually exclusive.

1. Introduction to Probability

Probability measures how likely an event is to occur. It ranges from 0 (impossible) to 1 (certain).

Key Definitions

• Experiment: A process with uncertain outcomes (e.g., rolling a die, checking if a Khalti payment succeeds).
• Sample Space (S): All possible outcomes (e.g., for a die: {1, 2, 3, 4, 5, 6}).
• Event (E): A subset of the sample space (e.g., rolling an even number: {2, 4, 6}).
• Probability of an Event: $$P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

Example 1: Probability of a Die Roll

Question: What is the probability of rolling a 4 on a fair six-sided die? Solution:

• Sample space $S=\{1,2,3,4,5,6\}$.
• Favorable outcome = {4}.
• $P(4)=\frac{1}{6}\approx 0.1667$ or 16.67%.

Example 2: Probability of a Khalti Payment Success

Scenario: Khalti reports that 95% of transactions succeed. What is the probability a random transaction fails? Solution:

• Probability of success $P(S)=0.95$.
• Probability of failure $P(F)=1-P(S)=1-0.95=0.05$ or 5%.

2. Types of Events

A. Mutually Exclusive (Disjoint) Events

• Two events cannot occur simultaneously.
• Example: Rolling a die and getting both 2 and 4 (impossible).
• Addition Rule: $$P(A\text{ or }B)=P(A)+P(B)\text{if }A\text{ and }B\text{ are mutually exclusive.}$$

B. Independent Events

• Occurrence of one event does not affect the other.
• Example: Flipping a coin and rolling a die (coin flip does not influence die roll).
• Multiplication Rule: $$P(A\text{ and }B)=P(A)\times P(B)\text{if }A\text{ and }B\text{ are independent.}$$

C. Conditional Probability

• Probability of an event given another event has occurred.
• Formula: $$P(A|B)=\frac{P(A\text{ and }B)}{P(B)}$$
• Example: Pathao knows 20% of rides are canceled. If 10% of canceled rides are due to traffic, what is the probability a ride is canceled because of traffic?
  • $P(\text{Canceled})=0.20$
  • $P(\text{Canceled and Traffic})=0.10$
  • $P(\text{Traffic}|\text{Canceled})=\frac{0.10}{0.20}=0.5$ or 50%.

3. Probability Rules

A. Addition Rule

• For any two events (mutually exclusive or not): $$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$$
• Example: In a class of 50 students, 20 like coffee, 15 like tea, and 5 like both. What is the probability a student likes coffee or tea?
  • $P(\text{Coffee})=\frac{20}{50}=0.4$
  • $P(\text{Tea})=\frac{15}{50}=0.3$
  • $P(\text{Coffee and Tea})=\frac{5}{50}=0.1$
  • $P(\text{Coffee or Tea})=0.4+0.3-0.1=0.6$ or 60%.

B. Multiplication Rule

• For independent events: $$P(A\text{ and }B)=P(A)\times P(B)$$
• Example: Ncell has a 90% chance of network uptime, and Khalti has a 98% chance of payment processing. What is the probability both work?
  • $P(\text{Ncell})=0.90$
  • $P(\text{Khalti})=0.98$
  • $P(\text{Both})=0.90\times 0.98=0.882$ or 88.2%.

C. Complement Rule

• Probability of an event not occurring: $$P(\text{Not }A)=1-P(A)$$
• Example: If Daraz delivers 92% of orders on time, what is the probability an order is delayed?
  • $P(\text{Delayed})=1-0.92=0.08$ or 8%.

4. Bayes’ Theorem

Used to update probabilities based on new information. $$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$$ Example: eSewa detects fraud in 5% of transactions. If 3% of all transactions are fraudulent, and 60% of fraudulent transactions are flagged, what is the probability a flagged transaction is actually fraudulent?

• $P(\text{Fraud})=0.05$
• $P(\text{Flagged}|\text{Fraud})=0.60$
• $P(\text{Flagged})=P(\text{Flagged}|\text{Fraud})\times P(\text{Fraud})+P(\text{Flagged}|\text{Not Fraud})\times P(\text{Not Fraud})$
  • Assume 1% of non-fraudulent transactions are flagged:
  • $P(\text{Flagged})=(0.60\times 0.05)+(0.01\times 0.95)=0.03+0.0095=0.0395$
• $P(\text{Fraud}|\text{Flagged})=\frac{0.60\times 0.05}{0.0395}\approx 0.76$ or 76%.

graph TD
    A["Prior Probability: P(Fraud) = 5%"] --> B["Likelihood: P(Flagged|Fraud) = 60%"]
    C["Prior Probability: P(Not Fraud) = 95%"] --> D["Likelihood: P(Flagged|Not Fraud) = 1%"]
    B & D --> E["Total P(Flagged) = 3.95%"]
    E --> F["Posterior: P(Fraud|Flagged) ≈ 76%"]

5. Probability Distributions

A. Discrete Probability Distributions

• Binomial Distribution: Models fixed number of trials (success/failure).
  • Formula: $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
  • Example: Ncell has a 10% chance of customer churn per month. What is the probability exactly 2 out of 5 customers churn?
    • $n=5$, $k=2$, $p=0.10$
    • $P(X=2)=\left(\genfrac{}{}{0ex}{}{5}{2}\right)(0.10{)}^2(0.90{)}^3=10\times 0.01\times 0.729=0.0729$ or 7.29%.

B. Continuous Probability Distributions

• Normal Distribution: Symmetric, bell-shaped curve (e.g., NEPSE stock returns).
  • Formula: $$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$$
  • Example: NEPSE stock returns have a mean (μ) of 0% and standard deviation (σ) of 10%. What is the probability returns are between -5% and +5%?
    • Use Z-score: $$Z=\frac{X-\mu }{\sigma }$$ For $X=-5\%$: $Z=\frac{-5-0}{10}=-0.5$ For $X=+5\%$: $Z=\frac{5-0}{10}=0.5$
    • From Z-table, $P(-0.5<Z<0.5)\approx 0.3829$ or 38.29%.

C. Poisson Distribution

• Models rare events (e.g., Daraz delivery delays).
  • Formula: $$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$$
  • Example: Daraz expects 3 delays per 100 orders. What is the probability of exactly 2 delays in 50 orders?
    • $\lambda =\frac{3}{100}\times 50=1.5$
    • $P(X=2)=\frac{e^{-1.5}(1.5{)}^2}{2!}\approx 0.251$ or 25.1%.

## In the Real World

1. Khalti & eSewa Fraud Detection

   • Idea Used: Bayes’ Theorem and Conditional Probability.
   • How: Khalti uses transaction history to calculate $P(\text{Fraud}|\text{Suspicious Activity})$. If a user suddenly transfers 50,000 NPR to an unknown account, the app flags it based on prior fraud patterns.

2. Ncell Customer Churn Prediction

   • Idea Used: Binomial Distribution.
   • How: Ncell models customer retention using binomial probability. If 10% of customers leave monthly, they predict 2 out of 20 customers will churn in a quarter using $P(X=2)=\left(\genfrac{}{}{0ex}{}{20}{2}\right)(0.10{)}^2(0.90{)}^{18}$.

3. Daraz Delivery Time Estimates

   • Idea Used: Poisson Distribution.
   • How: Daraz estimates delivery delays. If 2% of orders are delayed daily, they calculate $P(X=3\text{ delays in 100 orders})$ using Poisson to optimize logistics.

4. NEPSE Stock Risk Assessment

   • Idea Used: Normal Distribution.
   • How: Investors use the 68-95-99.7 rule (empirical rule) to assess risk. If NEPSE returns are normally distributed ($\mu =0\%$, $\sigma =10\%$), 68% of returns fall between -10% and +10%.

5. Bank Loan Default Risk (Global Ime)

   • Idea Used: Expected Value.
   • How: Global Ime calculates the expected loss from loan defaults. If a 10,000 NPR loan has a 5% default rate, the expected loss is $10,000\times 0.05=500$ NPR per loan.

## Exam Tip

1. Always define events clearly (e.g., "Let A = event of a Khalti payment failing").
2. Draw Venn diagrams or trees for problems involving multiple events (e.g., "A student likes coffee or tea").
3. Check independence: If events are not independent, use $P(A\text{ and }B)=P(A|B)\times P(B)$.
4. Memorize key formulas:
   • Addition: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
   • Multiplication: $P(A\cap B)=P(A)\times P(B)$ (if independent)
   • Bayes’: $P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$
5. For distributions:
   • Use binomial for fixed trials (e.g., Ncell churn).
   • Use Poisson for rare events (e.g., Daraz delays).
   • Use normal for symmetric data (e.g., NEPSE returns).
6. Units matter: Probabilities are unitless (always between 0 and 1), but expected value has units (e.g., NPR).

Practice Question: NTC reports that 80% of its internet connections are stable. If 15% of unstable connections are due to power cuts, and 10% of all connections are unstable:

1. What is the probability a connection is unstable due to a power cut?
2. If a connection is flagged as unstable, what is the probability it is not due to a power cut?

Solution:

1. $P(\text{Unstable})=0.10$ $P(\text{Power Cut}|\text{Unstable})=0.15$ $P(\text{Unstable and Power Cut})=0.15\times 0.10=0.015$ or 1.5%.
2. $P(\text{Not Power Cut}|\text{Unstable})=1-0.15=0.85$ or 85%.

In the real world

• Khalti Fraud Detection: Uses Bayes' Theorem to update the probability that a transaction is fraudulent based on new evidence (e.g., unusual location or amount), helping to flag suspicious activity with higher accuracy.
• Ncell Customer Churn: Uses the Binomial Distribution to model the probability of a specific number of customers leaving in a month, aiding in retention strategy planning.
• Daraz Delivery Logistics: Uses the Poisson Distribution to estimate the probability of a certain number of delivery delays per day, allowing for better resource allocation and customer expectation management.