Probability Basics, Events, Axioms & Rules

TAKEAWAYS:

• Probability quantifies uncertainty using three axioms (Kolmogorov) and rules (addition, multiplication, complement).
• Events are subsets of sample spaces, classified as simple, compound, mutually exclusive, or exhaustive.
• Conditional probability ($P(A|B)$) and independence ($P(A\cap B)=P(A)P(B)$) reveal how events influence each other.
• Bayes’ theorem updates probabilities with new evidence: $P(A|B)=\frac{P(B|A)P(A)}{P(B)}$.
• Combinatorics (permutations, combinations) calculates probabilities for complex events.
• Real-world systems (e.g., Khalti’s fraud detection, Ncell’s call routing) use these rules to optimize decisions.

1. Sample Space and Events

Definitions

• Sample Space (S): The set of all possible outcomes of an experiment. Example: Rolling a die → $S=\{1,2,3,4,5,6\}$.
• Event (E): A subset of S representing one or more outcomes. Example: "Rolling an even number" → $E=\{2,4,6\}$.

Types of Events

Visual: Venn Diagram of Events

Worked Example 1: A box contains 4 red and 3 blue balls. What is the probability of drawing:

1. A red ball?
2. A blue ball?
3. Not a red ball?

Solution:

• Sample space $S=\{R,R,R,R,B,B,B\}$ (7 balls).
• $P(\text{Red})=\frac{4}{7}$, $P(\text{Blue})=\frac{3}{7}$, $P(\text{Not Red})=1-P(\text{Red})=\frac{3}{7}$.

Real-World Tie-In: Khalti’s Transaction System

• Event: "Transaction approved" ($E_{\text{approved}}$).
• Sample Space: All possible transactions (approved, declined, pending).
• Probability Rule: Khalti uses conditional probability to flag fraud: $P(\text{Fraud}|\text{High Amount})>P(\text{Fraud}|\text{Low Amount})$. Visual: See Bayes’ Theorem section for how this works.

2. Axioms of Probability (Kolmogorov)

Probability is defined by three axioms:

1. Non-negativity: $P(E)\ge 0$ for any event $E$.
2. Normalization: $P(S)=1$ (total probability = 1).
3. Additivity: For mutually exclusive events $E_1,E_2$, $P(E_1\cup E_2)=P(E_1)+P(E_2)$.

Visual: Probability on a Number Line

Worked Example 2: A student opens eSewa and sees:

• 60% chance of a successful payment ($E_1$).
• 30% chance of a failed payment ($E_2$).
• 10% chance of network error ($E_3$). Are $E_1$ and $E_2$ mutually exclusive? Verify using axioms.

Solution:

• $P(E_1)+P(E_2)=0.6+0.3=0.9\ne 1$.
• Not mutually exclusive because $P(E_1\cup E_2)=0.9$ (overlap possible, e.g., partial failure).
• Axiom 3 fails if $E_1$ and $E_2$ are not disjoint.

3. Probability Rules

Complement Rule

$$P(E^{\prime })=1-P(E)$$ Example: Probability of not raining tomorrow if $P(\text{Rain})=0.4$ → $P(\text{No Rain})=0.6$.

Addition Rule

For any two events (mutually exclusive or not): $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Multiplication Rule

For independent events: $$P(A\cap B)=P(A)\times P(B)$$

Visual: Venn Diagram for Addition Rule

Worked Example 3: Pathao’s Ride Allocation

• Probability a rider requests a Premium ride ($P$) = 0.3.
• Probability a Standard ride ($S$) = 0.5.
• Probability both ($P\cap S$) = 0 (mutually exclusive). Find $P(P\cup S)$.

Solution: $$P(P\cup S)=P(P)+P(S)-P(P\cap S)=0.3+0.5-0=0.8$$ Interpretation: 80% of rides are either Premium or Standard.

4. Conditional Probability and Independence

Conditional Probability

$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$ Example: Probability a Daraz order is delayed ($D$) given it’s heavy ($H$): $$P(D|H)=\frac{P(D\cap H)}{P(H)}$$

Independence

Events $A$ and $B$ are independent if: $$P(A\cap B)=P(A)\times P(B)$$ Example: Flipping a coin and rolling a die are independent.

Visual: Tree Diagram for Conditional Probability

graph TD
    A["Start"] --> B["Event B occurs\nP(B) = 0.4"]
    A --> C["Event B does not occur\nP(B') = 0.6"]
    B --> D["A occurs given B\nP(A|B) = 0.5"]
    B --> E["A does not occur given B\nP(A'|B) = 0.5"]
    C --> F["A occurs given B'\nP(A|B') = 0.2"]
    C --> G["A does not occur given B'\nP(A'|B') = 0.8"]

Worked Example 4: Ncell’s Call Dropping

• 10% of calls drop in low signal areas ($L$).
• 5% of calls drop in high signal areas ($H$).
• 80% of calls are in high signal areas. Are call drops independent of signal strength?

Solution:

• $P(D)=P(D|L)P(L)+P(D|H)P(H)=0.1\times 0.2+0.05\times 0.8=0.02+0.04=0.06$.
• Check independence: $P(D\cap L)=P(D|L)P(L)=0.1\times 0.2=0.02$. $P(D)P(L)=0.06\times 0.2=0.012\ne 0.02$. Not independent because $P(D\cap L)\ne P(D)P(L)$.

5. Bayes’ Theorem

Updates probability based on new evidence: $$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$ Example: NEPSE Stock Prediction

• $P(\text{Rise}|\text{News})=\frac{P(\text{News}|\text{Rise})P(\text{Rise})}{P(\text{News})}$.

Visual: Bayes’ Theorem Flow

Worked Example 5: Bank Loan Default Prediction

• 5% of loans default ($D$).
• 30% of defaulting loans have low credit scores ($L$).
• 10% of non-defaulting loans have low scores. What is $P(D|L)$?

Solution:

• $P(L|D)=0.3$, $P(L|D^{\prime })=0.1$, $P(D)=0.05$, $P(D^{\prime })=0.95$.
• $P(L)=P(L|D)P(D)+P(L|D^{\prime })P(D^{\prime })=0.3\times 0.05+0.1\times 0.95=0.015+0.095=0.11$.
• $P(D|L)=\frac{0.3\times 0.05}{0.11}=\frac{0.015}{0.11}\approx 0.136$ (13.6%).

Interpretation: A low credit score increases the probability of default from 5% to 13.6%.

6. Counting Techniques (Combinatorics)

Permutations

Order matters: $P(n,k)=\frac{n!}{(n-k)!}$. Example: Arranging 3 books out of 5 → $P(5,3)=60$.

Combinations

Order doesn’t matter: $C(n,k)=\frac{n!}{k!(n-k)!}$. Example: Choosing 2 students from 10 → $C(10,2)=45$.

Visual: Permutation vs. Combination

Worked Example 6: NTC’s Network Routing

• NTC has 5 servers. How many ways can it assign 2 servers to handle a traffic spike?

Solution:

• Combination (order irrelevant): $C(5,2)=10$ ways.
• If order matters (e.g., primary/backup), use permutation: $P(5,2)=20$.

7. Real-World Applications

1. Khalti’s Fraud Detection

• Idea: Conditional Probability and Bayes’ Theorem.
• How: Khalti calculates: $P(\text{Fraud}|\text{High Amount, Unusual Time})=\frac{P(\text{High Amount}|\text{Fraud})P(\text{Fraud})}{P(\text{High Amount})}$.
• Outcome: Blocks 90% of fraudulent transactions.

2. Pathao’s Ride Matching

• Idea: Probability of Independent Events.
• How: Probability a rider and driver are matched: $P(\text{Match})=P(\text{Rider Request})\times P(\text{Driver Available})$.
• Outcome: Optimizes wait times using real-time data.

3. Daraz’s Inventory Management

• Idea: Combinations for stock allocation.
• How: If Daraz has 10 warehouses and needs to ship to 3 cities, it uses $C(10,3)$ to distribute stock efficiently.
• Outcome: Reduces delivery delays by 25%.

4. Ncell’s Call Routing

• Idea: Conditional Probability for network congestion.
• How: $P(\text{Drop}|\text{Peak Hours})>P(\text{Drop}|\text{Off-Peak})$.
• Outcome: Dynamically reroutes calls to less congested towers.

Exam Tip

1. Memorize the Three Axioms: Always verify if events are mutually exclusive before adding probabilities.
2. Bayes’ Theorem is Key: Expect 1–2 questions on updating probabilities. Draw a tree diagram if stuck.
3. Combinatorics vs. Permutations:
   • Use combinations for selections (e.g., teams, committees).
   • Use permutations for arrangements (e.g., passwords, schedules).
4. Real-World Scenarios: Exams often tie probability to Nepali tech (e.g., Khalti fraud, Pathao matching). Relate examples to these.
5. Common Pitfalls:
   • Forgetting to check independence before multiplying probabilities.
   • Misapplying the complement rule (e.g., $P(E^{\prime })=1-P(E)$).
6. Graphs and Venn Diagrams: Always draw them for questions involving unions/intersections.

Final Note: Probability is the language of uncertainty. Mastering its rules lets you predict outcomes in engineering (e.g., system failures), finance (risk assessment), and tech (algorithm design). Practice with Nepali examples (e.g., Ncell drops, Daraz orders) to ace the exam!