Classical or Mathematical Probability Examples

Data Science and A.I. Lecture Series

 

What You Will Learn

• The definition and basic concepts of probability.
• Examples of classical probability problems.
• Application of probability rules such as complements and odds.
• Step-by-step solutions to real-world probability problems.

Introduction

Probability is the study of uncertainty. It provides tools to measure the likelihood of events.

Key historical contributions:

• Galileo: Analyzed dice probabilities.
• Pascal and Fermat: Created the mathematical theory of probability.

Applications include games of chance, decision-making, and statistical inference.

Classical Probability Definition

Definition:

\[
P(A) = \frac{\text{Number of favorable cases}}{\text{Number of exhaustive cases}}, \quad \text{where } 0 \leq P(A) \leq 1.
\]

Complement Rule: \(P(A) + P(\overline{A}) = 1\).

Examples:

• Tossing a coin: \(P(\text{Head}) = 0.5\).
• Rolling a die: \(P(\text{Even number}) = \frac{3}{6}\).

Example 1: Tossing a Coin Twice

Problem: Find the probability of getting at least one head when two coins are tossed.

Sample Space: \(S = \{HH, HT, TH, TT\}\).

Event: At least one head \(E = \{HH, HT, TH\}\).

Solution: \(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{4}\).

Example 2: Rolling a Die

Problem 1: Find the probability of rolling a prime number.

Sample Space: \(S = \{1, 2, 3, 4, 5, 6\}\).

Event: Prime numbers \(E = \{2, 3, 5\}\).

Solution: \(P(E) = \frac{3}{6} = \frac{1}{2}\).

Problem 2: Find the probability of rolling a number greater than 4.

Event: \(E = \{5, 6\}\).

Solution: \(P(E) = \frac{2}{6} = \frac{1}{3}\).

Example 3: Drawing a Card

Problem 1: Find the probability of drawing a red card from a standard deck of 52 cards.

Total Cards: 52.

Red Cards: 26.

Solution: \(P(\text{Red}) = \frac{26}{52} = \frac{1}{2}\).

Problem 2: Find the probability of drawing a face card.

Face Cards: 12 (Jack, Queen, King in each suit).

Solution: \(P(\text{Face}) = \frac{12}{52} = \frac{3}{13}\).

Example 4: Events with Dice

Problem 1: Find the probability of getting a sum greater than 8 when two dice are thrown.

Total Outcomes: 36.

Favorable Outcomes: \(\{(3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6)\}\) (10 outcomes).

Solution: \(P(\text{Sum} > 8) = \frac{10}{36} = \frac{5}{18}\).

Problem 2: Find the probability of getting a doublet (same number on both dice).

Favorable Outcomes: \(\{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\}\).

Solution: \(P(\text{Doublet}) = \frac{6}{36} = \frac{1}{6}\).

Example 5: Odds and Probability

Problem: The odds in favor of an event \(A\) are 4:3. Find \(P(A)\).

Formula: \(P(A) = \frac{\text{Odds in favor of } A}{\text{Total odds}}\).

Solution: \(P(A) = \frac{4}{4+3} = \frac{4}{7}\).

PDF Presentation-

classicprobabilityexamples

Video

Summary

• Classical probability: \(P(A) = \frac{\text{Favorable Cases}}{\text{Exhaustive Cases}}\).
• Complementary Rule: \(P(A) + P(\overline{A}) = 1\).
• Odds measure favorability ratios.
• Practice enhances understanding.

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