Probability Theory: Rules, Events & Calculations

TAKEAWAYS:

• Understand probability basics: sample space, events, and their types (mutually exclusive, independent, complementary).
• Master addition and multiplication rules to calculate probabilities of combined events.
• Learn how to apply conditional probability in real-world scenarios like risk assessment or business forecasting.
• Differentiate between dependent and independent events and their impact on probability calculations.
• Use probability trees and Venn diagrams to visualize complex probability problems.
• Recognize how probability theory underpins decision-making in finance, logistics, and technology.

1. Basic Concepts of Probability

Probability is the measure of the likelihood that an event will occur. It ranges from 0 (impossible) to 1 (certain).

Key Definitions

• Experiment: A process that leads to one of several possible outcomes (e.g., rolling a die, flipping a coin).
• Sample Space (S): The set of all possible outcomes of an experiment.
• Event (E): A subset of the sample space (e.g., rolling an even number: E = {2, 4, 6}).
• Probability of an Event (P(E)): $$P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

Types of Events

1. Mutually Exclusive (Disjoint) Events: Two events cannot occur simultaneously. Example: Rolling a die and getting a 3 or a 5 (cannot happen at the same time).
2. Independent Events: The occurrence of one event does not affect the probability of another. Example: Flipping a coin twice (first flip does not influence the second).
3. Dependent Events: The occurrence of one event affects the probability of another. Example: Drawing two cards from a deck without replacement.
4. Complementary Events: Two events are complementary if one event is the negation of the other. Example: Getting a head or a tail when flipping a coin.

Worked Example 1: Basic Probability Calculation

Problem: A bag contains 5 red balls and 3 blue balls. What is the probability of drawing a blue ball? Solution:

• Total balls = 5 (red) + 3 (blue) = 8.
• Favorable outcomes (blue balls) = 3.
• Probability of drawing a blue ball: $$P(\text{Blue})=\frac{3}{8}=0.375\text{ or }37.5\%$$

2. Addition Rule of Probability

The addition rule is used to find the probability of either of two events occurring.

Formula

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

• If A and B are mutually exclusive, $P(A\cap B)=0$, so: $$P(A\cup B)=P(A)+P(B)$$

Worked Example 2: Addition Rule (Mutually Exclusive Events)

Problem: In a class of 40 students, 15 study Statistics, and 20 study Economics. If no student studies both, what is the probability that a randomly selected student studies either Statistics or Economics? Solution:

• $P(\text{Statistics})=\frac{15}{40}=0.375$
• $P(\text{Economics})=\frac{20}{40}=0.5$
• Since the events are mutually exclusive: $$P(\text{Statistics or Economics})=0.375+0.5=0.875\text{ or }87.5\%$$

Worked Example 3: Addition Rule (Non-Mutually Exclusive Events)

Problem: In the same class, 5 students study both Statistics and Economics. What is the probability that a student studies either Statistics or Economics? Solution:

• $P(\text{Statistics}\cap \text{Economics})=\frac{5}{40}=0.125$
• Using the addition rule: $$P(\text{Statistics or Economics})=0.375+0.5-0.125=0.75\text{ or }75\%$$

3. Multiplication Rule of Probability

The multiplication rule is used to find the probability of both two events occurring.

Formula

For any two events A and B: $$P(A\cap B)=P(A)\times P(B|A)$$

• If A and B are independent, $P(B|A)=P(B)$, so: $$P(A\cap B)=P(A)\times P(B)$$

Worked Example 4: Multiplication Rule (Independent Events)

Problem: A company’s quality control checks show that 2% of its products are defective. If two products are selected at random, what is the probability that both are defective? Solution:

• $P(\text{Defective})=0.02$
• Since the events are independent: $$P(\text{Both Defective})=0.02\times 0.02=0.0004\text{ or }0.04\%$$

Worked Example 5: Multiplication Rule (Dependent Events)

Problem: A bag contains 4 red and 6 blue balls. Two balls are drawn without replacement. What is the probability that both are red? Solution:

• First draw: $P(\text{Red})=\frac{4}{10}=0.4$
• Second draw (without replacement): $P(\text{Red again})=\frac{3}{9}\approx 0.333$
• Combined probability: $$P(\text{Both Red})=0.4\times 0.333\approx 0.133\text{ or }13.3\%$$

4. Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred.

Formula

$$P(B|A)=\frac{P(A\cap B)}{P(A)}$$ Where:

• $P(B|A)$: Probability of B given A.
• $P(A\cap B)$: Probability of both A and B occurring.

Worked Example 6: Conditional Probability

Problem: In a factory, 10% of products are defective. If 80% of defective products are rejected in inspection, what is the probability that a randomly selected product is defective given that it was rejected? Solution:

• Let:
  • $P(D)=0.10$ (Probability of defective)
  • $P(R|D)=0.80$ (Probability of rejection given defective)
• Assume 5% of non-defective products are also rejected (for simplicity):
  • $P(R|\text{Not }D)=0.05$
• Total probability of rejection: $$P(R)=P(R|D)\times P(D)+P(R|\text{Not }D)\times P(\text{Not }D)=0.8\times 0.1+0.05\times 0.9=0.08+0.045=0.125$$
• Probability of defective given rejected: $$P(D|R)=\frac{P(R|D)\times P(D)}{P(R)}=\frac{0.8\times 0.1}{0.125}=\frac{0.08}{0.125}=0.64\text{ or }64\%$$

In the Real World

Probability theory is used in decision-making, risk assessment, and predictive modeling in many industries:

1. eSewa (Nepal):

   • Uses probability to estimate the likelihood of fraudulent transactions in digital payments.
   • Example: If 0.5% of transactions are fraudulent, eSewa’s algorithm calculates the probability that a flagged transaction is fraudulent using conditional probability.

2. Pathao (Ride-Hailing App):

   • Uses probability models to predict driver availability in different zones.
   • Example: If 60% of drivers are available in Kathmandu’s Thapathali area at 8 PM, Pathao’s algorithm multiplies this probability by the likelihood of a rider requesting a ride to calculate matching efficiency.

3. Nepal Rastra Bank (NRB) Loan Approvals:

   • Banks use probability of default to decide loan eligibility.
   • Example: If a customer’s credit score suggests a 15% chance of default, NRB’s risk model applies the multiplication rule to combine this with income stability probability before approving a loan.

4. NTC and Ncell Network Reliability:

   • Telecommunications companies use probability distributions to predict network congestion.
   • Example: If 70% of users in a tower area are active during peak hours, Ncell calculates the probability of call drops using dependent event probabilities (e.g., multiple users trying to connect simultaneously).

5. Daraz (E-Commerce) Order Fulfillment:

   • Daraz uses probability trees to estimate delivery delays.
   • Example: If 30% of orders face warehouse delays and 20% face shipping delays (independent events), the probability of a delay is calculated as: $$P(\text{Delay})=1-P(\text{No Delay})=1-(0.7\times 0.8)=1-0.56=0.44\text{ or }44\%$$

Visualizing Probability Concepts

Figure 1: Sample Space for Rolling a Die

Caption: The sample space $S$ for rolling a die includes all possible outcomes. Event $E$ (rolling an even number) is a subset of $S$.

Figure 2: Venn Diagram for Addition Rule (Non-Mutually Exclusive Events)

Caption: This Venn diagram shows two overlapping events $A$ and $B$. The intersection $A\cap B$ is subtracted to avoid double-counting.

Figure 3: Probability Tree for Daraz Order Delays

graph TD
    A["Start"] --> B["No Warehouse Delay (70%)"]
    A --> C["Warehouse Delay (30%)"]
    B --> D["No Shipping Delay (80%)"]
    B --> E["Shipping Delay (20%)"]
    C --> F["No Shipping Delay (80%)"]
    C --> G["Shipping Delay (20%)"]
    D --> H["No Delay (70% × 80% = 56%)"]
    E --> I["Delay (70% × 20% = 14%)"]
    F --> J["Delay (30% × 80% = 24%)"]
    G --> K["Delay (30% × 20% = 6%)"]

Caption: This probability tree shows the likelihood of order delays at Daraz. The final probabilities are calculated by multiplying branch probabilities.

Figure 4: Conditional Probability in eSewa Fraud Detection

Caption: This table shows the probability of a transaction being fraudulent given that it was flagged. $P(F|G)=\frac{0.004}{0.1}=0.04$ or 4%.

Exam Tip

1. Understand the difference between addition and multiplication rules:

   • Use addition for "OR" scenarios (e.g., "probability of A or B").
   • Use multiplication for "AND" scenarios (e.g., "probability of A and B").

2. Check for independence:

   • If events are independent, $P(A\cap B)=P(A)\times P(B)$.
   • If dependent, use $P(B|A)$.

3. Draw diagrams:

   • Venn diagrams for addition rules.
   • Probability trees for sequential events.

4. Common mistakes to avoid:

   • Forgetting to subtract $P(A\cap B)$ in the addition rule for non-mutually exclusive events.
   • Misapplying conditional probability (e.g., confusing $P(A|B)$ with $P(B|A)$).

5. Real-world applications:

   • Expect questions linking probability to business decisions (e.g., risk assessment, quality control).
   • Example: "A bank approves 60% of loan applications. If 10% of approved loans default, what is the probability a randomly selected loan is approved and defaults?"

Summary Table: Probability Rules

This note covers all key concepts, worked examples, and real-world applications to ensure full marks in your exam. Practice drawing Venn diagrams and probability trees to visualize problems!