Probability: Events, Rules, and Business Applications

TAKEAWAYS:

• Probability measures how likely an event is, from 0 (impossible) to 1 (certain), using the formula $P(E)=\frac{\text{number of favorable outcomes}}{\text{total possible outcomes}}$.
• Understand independent (one event does not affect another) and dependent events (one event changes the probability of another).
• Use the Addition Rule for "OR" events ($P(A\cup B)=P(A)+P(B)-P(A\cap B)$) and the Multiplication Rule for "AND" events ($P(A\cap B)=P(A)\times P(B)$).
• Conditional probability ($P(A|B)$) helps solve real-world problems like business risks or market trends.
• Apply probability to decision-making in business, such as calculating expected profit or loss.
• Practice NEB-style questions to avoid common mistakes like misapplying rules or ignoring sample space.

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1. Introduction to Probability

Probability is the branch of mathematics that deals with chances or likelihood of events happening. It helps us make informed decisions in business, finance, and everyday life.

Key Terms

• Experiment: Any activity with uncertain outcomes (e.g., rolling a die, flipping a coin).
• Sample Space (S): All possible outcomes of an experiment.
  • Example: Rolling a die → $S=\{1,2,3,4,5,6\}$.
• Event (E): A subset of the sample space (e.g., getting an even number when rolling a die).
• Probability of an Event (P(E)): The chance of an event happening, calculated as: $$P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

Example 1: Basic Probability

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: $$P(\text{Blue})=\frac{3}{8}=0.375\text{ or }37.5\%.$$

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2. Types of Events

A. Simple and Compound Events

• Simple Event: An event with a single outcome (e.g., rolling a 4 on a die).
• Compound Event: An event with multiple outcomes (e.g., rolling an even number on a die).

B. Independent and Dependent Events

• Independent Events: The outcome of one event does not affect the other.
  • Example: Flipping a coin twice (first flip does not affect the second).
• Dependent Events: The outcome of one event affects the other.
  • Example: Drawing two cards from a deck without replacement.

Example 2: Independent vs. Dependent Events

Problem:

1. What is the probability of getting two heads in a row when flipping a fair coin?
2. What is the probability of drawing two kings in a row from a standard deck of 52 cards?

Solution:

1. Independent Events (Coin Flips):

   • $P(\text{First Head})=\frac{1}{2}$.
   • $P(\text{Second Head})=\frac{1}{2}$.
   • Combined probability: $$P(\text{Two Heads})=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}.$$

2. Dependent Events (Drawing Kings):

   • First king: $\frac{4}{52}$.
   • Second king (without replacement): $\frac{3}{51}$.
   • Combined probability: $$P(\text{Two Kings})=\frac{4}{52}\times \frac{3}{51}=\frac{12}{2652}=\frac{1}{221}.$$

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3. Probability Rules

A. Addition Rule (OR Rule)

Used when two events cannot happen at the same time (mutually exclusive) or when they can. $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

• If $A$ and $B$ are mutually exclusive, $P(A\cap B)=0$.

Example 3: Addition Rule

Problem: A die is rolled. What is the probability of getting a 2 or an even number? Solution:

• $P(2)=\frac{1}{6}$.
• $P(\text{Even})=P(2,4,6)=\frac{3}{6}=\frac{1}{2}$.
• $P(2\cap \text{Even})=P(2)=\frac{1}{6}$ (since 2 is already even).
• Using the addition rule: $$P(2\cup \text{Even})=\frac{1}{6}+\frac{1}{2}-\frac{1}{6}=\frac{1}{2}.$$

B. Multiplication Rule (AND Rule)

Used when two events must both happen. $$P(A\cap B)=P(A)\times P(B)$$

• If events are dependent, use conditional probability.

Example 4: Multiplication Rule

Problem: A box has 3 red and 2 blue marbles. What is the probability of drawing a red marble and then a blue marble without replacement? Solution:

• $P(\text{First Red})=\frac{3}{5}$.
• After removing one red, remaining marbles = 4 (2 red, 2 blue).
• $P(\text{Second Blue})=\frac{2}{4}=\frac{1}{2}$.
• Combined probability: $$P(\text{Red then Blue})=\frac{3}{5}\times \frac{1}{2}=\frac{3}{10}.$$

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4. Conditional Probability

Conditional probability finds the probability of an event given that another event has already occurred. $$P(A|B)=\frac{P(A\cap B)}{P(B)}$$

Example 5: Conditional Probability

Problem: In a class of 30 students, 15 are girls and 10 like math. If 5 girls like math, what is the probability that a student who likes math is a girl? Solution:

• $P(\text{Girl})=\frac{15}{30}=\frac{1}{2}$.
• $P(\text{Likes Math})=\frac{10}{30}=\frac{1}{3}$.
• $P(\text{Girl and Likes Math})=\frac{5}{30}=\frac{1}{6}$.
• Using conditional probability: $$P(\text{Girl}|\text{Likes Math})=\frac{\frac{1}{6}}{\frac{1}{3}}=\frac{1}{2}.$$

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5. Applications of Probability in Business

Probability helps businesses in:

1. Risk Assessment: Calculating the chance of losses (e.g., insurance companies).
2. Market Forecasting: Predicting sales trends.
3. Decision Making: Choosing between risky and safe investments.
4. Quality Control: Estimating defective products in manufacturing.

Example 6: Business Application (Expected Profit)

Problem: A shopkeeper sells a product with:

• Probability of selling 10 units = 0.4 (profit = Rs. 500).
• Probability of selling 15 units = 0.6 (profit = Rs. 800). What is the expected profit? Solution: $$\text{Expected Profit}=(0.4\times 500)+(0.6\times 800)=200+480=\text{Rs. }680.$$

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6. Common Mistakes to Avoid

1. Ignoring Sample Space: Always list all possible outcomes.
2. Misapplying Rules: Use addition for "OR" and multiplication for "AND."
3. Forgetting Dependence: If events are dependent, adjust probabilities accordingly.
4. Incorrect Fractions: Simplify fractions properly (e.g., $\frac{2}{4}=\frac{1}{2}$).

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Exam Tip

• NEB Exam Pattern: Expect questions on:
  • Calculating basic probabilities.
  • Applying addition/multiplication rules.
  • Solving conditional probability problems.
  • Real-world business applications (e.g., expected value).
• Marks Distribution:
  • 2–4 marks for simple probability calculations.
  • 5–7 marks for applying rules to word problems.
  • 8–10 marks for business-related probability questions.
• Key Formulas to Memorize:
  • $P(E)=\frac{\text{Favorable}}{\text{Total}}$.
  • $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
  • $P(A\cap B)=P(A)\times P(B)$ (independent events).
  • $P(A|B)=\frac{P(A\cap B)}{P(B)}$.

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NEB-Style Practice Questions

Short Answer (2–4 marks)

1. A bag has 4 red and 6 blue balls. What is the probability of drawing a red ball?
2. If two events $A$ and $B$ are mutually exclusive, what is $P(A\cup B)$?

Long Answer (5–10 marks)

3. A company has two machines:
   • Machine A produces 60% of items, with 2% defective.
   • Machine B produces 40%, with 5% defective. If a defective item is found, what is the probability it came from Machine A?
4. A die is rolled twice. What is the probability of getting a sum of 7?

Business Application (8–10 marks)

5. A shopkeeper buys a product at Rs. 100. He sells it at Rs. 150 with a 30% chance of selling 10 units or Rs. 200 with a 70% chance of selling 5 units. Calculate the expected profit.

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Answers:

1. $\frac{4}{10}=\frac{2}{5}$.
2. $P(A\cup B)=P(A)+P(B)$.
3. Use Bayes’ Theorem: $P(A|\text{Defective})=\frac{0.6\times 0.02}{0.6\times 0.02+0.4\times 0.05}$.
4. Possible pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → $\frac{6}{36}=\frac{1}{6}$.
5. Expected profit = $(0.3\times (150-100)\times 10)+(0.7\times (200-100)\times 5)=\text{Rs. }5100$.