Analyzing Probabilities Using Contingency Tables

Analyzing Probabilities Using Contingency Tables

Introduction to Contingency Tables

Contingency tables are a fundamental tool in statistics for summarizing the relationship between two or more categorical variables. They allow us to compute various types of probabilities, including marginal, joint, and conditional probabilities, which are essential for understanding associations between variables.

Types of Probabilities from Contingency Tables

• Marginal Probability: The probability of an event occurring in a single category, regardless of other variables.

• Joint Probability: The probability of two events occurring together (i.e., the intersection of two categories).

• Conditional Probability: The probability of one event occurring given that another event has already occurred.

Key Formulas

• Marginal Probability: $P(A) = \frac{\text{Number in category A}}{\text{Total number}}$

• Joint Probability: $P(A \cap B) = \frac{\text{Number in both A and B}}{\text{Total number}}$

• Conditional Probability: $P(A|B) = \frac{\text{Number in both A and B}}{\text{Number in category B}}$

Example: High School Student Survey

The following contingency table summarizes the results from a survey of 100 high school students regarding whether they drive a car and whether they own a cell phone.

	Drives a Car	Does Not Drive	Total
Owns Cell Phone	60	25	85
Does Not Own Cell Phone	5	10	15
Total	65	35	100

• Marginal Probability Example: Probability that a student drives a car: $P(\text{Drives a car}) = \frac{65}{100} = 0.65$

• Joint Probability Example: Probability that a student owns a cell phone and drives a car: $P(\text{Owns cell phone} \cap \text{Drives a car}) = \frac{60}{100} = 0.60$

• Conditional Probability Example: Probability that a student drives a car given they own a cell phone: $P(\text{Drives a car} | \text{Owns cell phone}) = \frac{60}{85} \approx 0.706$

Example: Drug Trial for ADHD Medication

This table shows the results from a drug trial for a new ADHD medication. Probabilities can be calculated for different outcomes and groups.

	Placebo	Non-Placebo	Total
Improved	10	40	50
Not Improved	40	10	50
Total	50	50	100

• Conditional Probability Example: Probability that a person's symptoms improved given they received the placebo: $P(\text{Improved} | \text{Placebo}) = \frac{10}{50} = 0.20$

• Joint Probability Example: Probability that a person's symptoms did not improve and they received the non-placebo: $P(\text{Not Improved} \cap \text{Non-Placebo}) = \frac{10}{100} = 0.10$

• Marginal Probability Example: Probability that a person's symptoms improved: $P(\text{Improved}) = \frac{50}{100} = 0.50$

Constructing Contingency Tables from Data

Contingency tables can be constructed from raw survey data by organizing counts according to categories. This helps in visualizing and calculating probabilities.

• Example: Suppose 50 people were surveyed about eye color and hair color. The table below summarizes the results.

	Black	Brown	Blond	Total
Blue	3	7	5	15
Brown	6	8	1	15
Hazel	4	9	7	20
Total	13	24	13	50

Conditional and Marginal Distributions

Conditional distributions show the distribution of one variable for a fixed value of another variable. Marginal distributions show the totals for each category of a single variable.

• Example: The table below shows the results from a survey of guests at a wedding for the catering menu. Find the conditional distribution for vegetarians and the marginal distribution of diet types.

	Vegetarian	Vegan	Regular	Total
Yes	78	16	6	100
No	2	3	15	20
Total	80	19	21	120

• Conditional Distribution Example: Among vegetarians, the proportion who are vegan is $\frac{16}{78} \approx 0.205$.

• Marginal Distribution Example: The proportion of guests who are vegetarian: $\frac{80}{120} = 0.667$.

Summary Table: Types of Probabilities in Contingency Tables

Type	Definition	Formula	Example
Marginal	Probability of a single event/category	$P(A) = \frac{\text{Total in A}}{\text{Grand Total}}$	Probability a student drives a car
Joint	Probability of two events occurring together	$P(A \cap B) = \frac{\text{Number in both A and B}}{\text{Grand Total}}$	Probability a student owns a cell phone and drives a car
Conditional	Probability of one event given another	$P(A|B) = \frac{\text{Number in both A and B}}{\text{Total in B}}$	Probability a student drives a car given they own a cell phone

Additional info: These notes expand on the brief examples and tables in the original material, providing definitions, formulas, and context for each type of probability and how to interpret contingency tables in statistical analysis.