Learning Objectives

By the end of this chapter, you will be able to:

  • Explain and apply the classical approach to probability
  • Calculate probabilities using the relative frequency approach
  • Understand when to use subjective probability
  • Compare the three approaches and their limitations
  • Solve probability problems using appropriate methods

What is Probability?

Probability is a numerical measure of the likelihood that an event will occur.

\[0 \leq P(A) \leq 1\] 
flowchart LR
    A["P(A) = 0<br/>Impossible"] --> B["P(A) = 0.5<br/>Equally Likely"] --> C["P(A) = 1<br/>Certain"]

    D["No chance"] --> E["50-50 chance"] --> F["Guaranteed"]
Probability Value Interpretation
P(A) = 0 Event A is impossible
P(A) = 0.25 25% chance, unlikely
P(A) = 0.50 50% chance, equally likely
P(A) = 0.75 75% chance, likely
P(A) = 1 Event A is certain

Three Approaches to Probability

flowchart TD
    A[Approaches to Probability] --> B[1. Classical<br/>A Priori]
    A --> C[2. Relative Frequency<br/>Empirical]
    A --> D[3. Subjective<br/>Personal]

    B --> B1["Based on logical analysis<br/>Before experiment"]
    C --> C1["Based on observed data<br/>After experiment"]
    D --> D1["Based on expert judgment<br/>Personal belief"]

1. Classical (A Priori) Approach

Definition

If an experiment has n equally likely outcomes and m of them are favorable to event A:

\[P(A) = \frac{m}{n} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]

Assumptions

  1. All outcomes are equally likely
  2. Total outcomes are finite and known
  3. Based on logical reasoning, not actual trials

Example 1: Coin Toss

Problem: What is the probability of getting a Head?

Solution:

  • Total outcomes (n) = 2 {H, T}
  • Favorable outcomes (m) = 1 {H}
\[P(\text{Head}) = \frac{1}{2} = 0.5\]

Example 2: Rolling a Die

Problem: Find the probability of getting: (a) A 4 (b) An even number (c) A number greater than 2

Solution:

Sample space S = {1, 2, 3, 4, 5, 6}, n = 6

(a) Getting a 4:

  • Favorable outcomes = {4}, m = 1
\[P(4) = \frac{1}{6} = 0.167\]

(b) Even number:

  • Favorable outcomes = {2, 4, 6}, m = 3
\[P(\text{even}) = \frac{3}{6} = \frac{1}{2} = 0.5\]

(c) Number greater than 2:

  • Favorable outcomes = {3, 4, 5, 6}, m = 4
\[P(>2) = \frac{4}{6} = \frac{2}{3} = 0.667\]

Example 3: Playing Cards

A standard deck has 52 cards:

  • 4 suits: ♠ Spades, ♥ Hearts, ♦ Diamonds, ♣ Clubs
  • Each suit has 13 cards: A, 2-10, J, Q, K
  • 26 red cards (hearts, diamonds), 26 black cards

Problem: Find the probability of drawing: (a) A King (b) A Heart (c) A Red card

Solution:

(a) King: 4 kings in deck

\[P(\text{King}) = \frac{4}{52} = \frac{1}{13} = 0.077\]

(b) Heart: 13 hearts in deck

\[P(\text{Heart}) = \frac{13}{52} = \frac{1}{4} = 0.25\]

(c) Red card: 26 red cards

\[P(\text{Red}) = \frac{26}{52} = \frac{1}{2} = 0.5\]

Limitations of Classical Approach

  1. Requires equally likely outcomes - not always realistic
  2. Must know all outcomes - may be impossible in complex situations
  3. Cannot handle continuous outcomes - only discrete events
  4. Ignores real-world data - based purely on theory

2. Relative Frequency (Empirical) Approach

Definition

Probability based on observed data from repeated experiments:

\[P(A) = \frac{\text{Number of times A occurred}}{\text{Total number of trials}} = \frac{f}{n}\]

As $n \to \infty$, this approaches the true probability (Law of Large Numbers).

When to Use

  • Outcomes are NOT equally likely
  • Historical data is available
  • Theory alone cannot determine probability

Example 4: Employee Absenteeism

Problem: Records show that out of 250 working days, an employee was absent 15 days. What is the probability of absence on any given day?

Solution:

\[P(\text{Absent}) = \frac{15}{250} = 0.06 = 6\%\]

Example 5: Quality Control

Problem: A factory inspected 500 products and found 35 defective. What is the probability that a randomly selected product is defective?

Solution:

\[P(\text{Defective}) = \frac{35}{500} = 0.07 = 7\%\]

Example 6: Election Prediction

Problem: A survey of 1,000 voters showed 420 support Candidate A. What is the probability a random voter supports Candidate A?

Solution:

\[P(\text{Support A}) = \frac{420}{1000} = 0.42 = 42\%\]

Advantages

  1. Based on actual observations
  2. Works when outcomes aren’t equally likely
  3. Can be updated with new data
  4. Practical for real-world applications

Limitations

  1. Requires large sample for accuracy
  2. Past may not predict future
  3. Sample may not be representative
  4. Cannot use before collecting data

3. Subjective Approach

Definition

Probability based on personal belief, judgment, or experience when:

  • No historical data exists
  • Event is unique/one-time
  • Expert opinion is the best available information

Formula

\[P(A) = \text{Personal assessment of likelihood}\]

Examples in Public Administration

Scenario Subjective Probability
New policy will succeed “70% confident”
Budget will be approved “Very likely - 85%”
Project completion on time “60% chance”
Natural disaster next year “Low - 20%”

Example 7: Policy Success

Problem: A policy expert believes there’s a 75% chance that a new education policy will improve literacy rates. What approach is being used?

Answer: This is a subjective probability based on the expert’s judgment, experience, and analysis. There’s no historical data for this exact policy.

When Subjective Probability is Used

flowchart TD
    A[Is historical data available?]
    A -->|Yes| B[Use Relative<br/>Frequency]
    A -->|No| C[Are outcomes<br/>equally likely?]
    C -->|Yes| D[Use Classical<br/>Approach]
    C -->|No| E[Use Subjective<br/>Approach]

    E --> F["Based on:<br/>- Expert opinion<br/>- Similar situations<br/>- Logical reasoning"]

Limitations

  1. Highly dependent on individual judgment
  2. May be biased
  3. Different experts may give different probabilities
  4. Difficult to verify accuracy

Comparison of Three Approaches

Aspect Classical Relative Frequency Subjective
Basis Logical reasoning Observed data Personal judgment
Timing Before experiment After experiment Before experiment
Requirements Equally likely outcomes Historical data Expert knowledge
Objectivity Highly objective Objective Subjective
Best For Games, simple events Repeated events Unique events
Example Coin toss, dice Quality control Policy success

Step-by-Step Example 8: Choosing the Right Approach

Problem: Identify the most appropriate probability approach for each situation:

(a) Probability of getting a sum of 7 when rolling two dice

Answer: Classical approach - outcomes are equally likely, can be calculated theoretically.

(b) Probability of machine breakdown based on last year’s records

Answer: Relative frequency approach - based on historical data.

(c) Probability of a newly proposed law being passed

Answer: Subjective approach - unique event, no historical data for this specific law.

(d) Probability of drawing a red ball from a bag with 5 red and 3 blue balls

Answer: Classical approach - outcomes can be enumerated and are equally likely.

Properties of Probability

Regardless of approach, all probabilities must satisfy these rules:

Axiom 1: Non-negativity

\[P(A) \geq 0\]

Axiom 2: Certainty

\[P(S) = 1\]

(Probability of sample space is 1)

Axiom 3: Additivity

For mutually exclusive events:

\[P(A \cup B) = P(A) + P(B)\]

Derived Properties

Complement Rule:

\[P(A') = 1 - P(A)\]

Example: If probability of rain is 0.3, probability of no rain is:

\[P(\text{No rain}) = 1 - 0.3 = 0.7\]

Practice Problems

Problem 1

A bag contains 6 white, 4 red, and 5 blue balls. A ball is drawn at random. Find the probability that it is: (a) White (b) Not red (c) Either white or blue

Problem 2

Hospital records show that out of 1,200 patients admitted last month, 180 had respiratory issues. What is the probability that a newly admitted patient has respiratory issues?

Problem 3

For each scenario, identify the probability approach: (a) Chance of winning a lottery with known ticket distribution (b) Probability of successful implementation of a new IT system (c) Machine failure rate based on 5 years of maintenance logs

Problem 4

A political analyst says there’s an 80% chance the ruling party will win the election. Is this a valid probability? What approach is being used?

Problem 5

If P(A) = 0.4, find: (a) P(A’) (b) Can P(A) + P(A’) ever exceed 1? Why or why not?

Summary

Approach Formula When to Use
Classical $P(A) = \frac{m}{n}$ Equally likely outcomes, games of chance
Relative Frequency $P(A) = \frac{f}{n}$ Historical data available
Subjective Personal judgment Unique events, expert opinion needed
Complement $P(A’) = 1 - P(A)$ Finding probability of “not A”

Next Topic

In the next chapter, we will study the Addition Rule of Probability - how to calculate probabilities when events are combined using OR (union).