Maths Literacy Matric Revision: Contingency tables

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Revision Notes: Probability – Contingency Tables for CAPS Mathematical Literacy Grade 12

Introduction

Probability is a branch of mathematics that deals with the likelihood or chance of different outcomes occurring. The use of contingency tables is a practical way to organize data and calculate probabilities. In CAPS Grade 12 Mathematical Literacy, understanding how to use contingency tables is essential for solving real-world problems that involve probability.

Key Points

  1. Basic Concepts:
  2. Probability: A measure of how likely an event is to occur.

  3. Contingency Table: A table used to display the frequency distribution of variables.

  4. Calculation Methods:

  5. Theoretical Probability: P(Event) = Number of favorable outcomes / Total number of possible outcomes.

  6. Experimental Probability: P(Event) = Number of times event occurs / Number of trials.

  7. Terminology:

  8. Events: Outcomes or occurrences that can be measured.

  9. Sample Space: Set of all possible outcomes.

  10. Contingency Table Components:

  11. Rows and Columns representing different categories or variables.

  12. Cells indicating the frequency count of occurrences.

  13. Probability Rules:

  14. Addition Rule: For mutually exclusive events: P(A or B) = P(A) + P(B).
  15. Multiplication Rule: For independent events: P(A and B) = P(A) × P(B).

Real-World Applications

Example Problem:
A contingency table is given below showing the smoking habits of men and women:

| Smoking Habits | Male | Female | Total |
|—————–|——|——–|——-|
| Smokes | 104 | 96 | 200 |
| Never Smokes | 98 | 10 | 108 |
| Social Smoker | 30 | 12 | 42 |
| Total | 232 | 118 | 350 |

Questions:
1. What is the probability that a randomly chosen male is a smoker?
– P(Male Smoker) = Number of male smokers / Total number of males
– P(Male Smoker) = 104 / 232 ≈ 0.45 or 45%

  1. What is the probability that a randomly chosen individual is a social smoker?
  2. P(Social Smoker) = Number of social smokers / Total individuals
  3. P(Social Smoker) = 42 / 350 ≈ 0.12 or 12%

Step-by-Step Solution:
1. Identify the relevant values from the table.
2. Substitute the values into the probability formula.
3. Simplify the fraction to get the decimal or percentage form of the probability.

Common Misconceptions and Errors

  1. Total Misinterpretation:

  2. Mistaking the sum of probabilities for mutually exclusive events should always be 1.

  3. Cell Value Errors:

  4. Confusing row totals with column totals can lead to incorrect probabilities.

  5. Independent vs. Mutually Exclusive Events:

  6. Ensure to differentiate between independent events and mutually exclusive events when applying the addition or multiplication rule.

Practice and Review

Practice Questions:
1. If a person is selected at random, what is the probability that the person is female and a social smoker?
– Practice Solution: P(Female and Social Smoker) = 12 / 350

  1. What is the probability of selecting a male who never smokes?
  2. Practice Solution: P(Male who never smokes) = 98 / 232

Examination Tips:
– Keywords: Look for phrases such as “mutually exclusive,” “independent events,” and specific categories stated in the questions.
– Manage your time: Practice solving similar problems within a fixed time to improve your speed and accuracy.

Connections and Extensions

  • Links to Other Topics:
  • Statistics: Contingency tables are crucial in organizing data for statistical analysis.

  • Data Handling: Understanding frequency distributions aids in interpreting and displaying data efficiently.

  • Real-World Implications:

  • Health Studies: Analyzing data related to smoking, dietary habits, etc.
  • Market Research: Decision-making based on consumer behavior.

Summary and Quick Review

  • Key Formulas:
  • P(Event) = Favorable Outcomes / Total Outcomes
  • Addition Rule for Mutually Exclusive Events

  • Multiplication Rule for Independent Events

  • Quick Reference:

  • Random Selection Probabilities
  • Contingency Table Analysis Steps

Additional Resources

  • Online educational platforms (e.g., Khan Academy).
  • Mathematics textbooks focusing on probability and statistics.
  • Interactive simulations and examples on educational websites.

This structured approach to understanding contingency tables in probability will equip students with the necessary skills and confidence to excel in their Grade 12 Mathematical Literacy exams.

References:
– Adapted from “Study & Master Mathematical Literacy Grade 12 Study Guide” .