Two-Way Tables Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

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A two-way table shows: Smoker/Cancer=30, Smoker/No-Cancer=70, Non-smoker/Cancer=20, Non-smoker/No-Cancer=180. Calculate marginal and joint proportions, and find P(Cancer|Smoker) vs P(Cancer|Non-smoker).

Solution

  1. 1
    Total: 300 people; Smokers=100, Non-smokers=200
  2. 2
    Joint: P(Smoker∩Cancer)=30/300=0.10P(\text{Smoker} \cap \text{Cancer}) = 30/300 = 0.10
  3. 3
    Marginal: P(Cancer)=50/300=0.167P(\text{Cancer}) = 50/300 = 0.167; P(Smoker)=100/300=0.333P(\text{Smoker}) = 100/300 = 0.333
  4. 4
    P(Cancer∣Smoker)=30/100=0.30P(\text{Cancer}|\text{Smoker}) = 30/100 = 0.30; P(Cancer∣Non-smoker)=20/200=0.10P(\text{Cancer}|\text{Non-smoker}) = 20/200 = 0.10

Answer

P(Cancer∣Smoker)=0.30P(\text{Cancer}|\text{Smoker}) = 0.30 vs. P(Cancer∣Non-smoker)=0.10P(\text{Cancer}|\text{Non-smoker}) = 0.10. Smokers have 3× higher cancer rate.
Two-way tables organize joint and conditional probabilities. Marginal probabilities are row/column totals divided by grand total. Conditional probabilities are cell frequencies divided by row (or column) totals. Comparing P(Cancer|Smoker) to P(Cancer|Non-smoker) reveals the association.

About Two-Way Tables

A table that displays frequencies for two categorical variables simultaneously, organized with one variable in rows and the other in columns. It shows joint frequencies (individual cells), marginal frequencies (row/column totals), and enables calculation of conditional frequencies.

Learn more about Two-Way Tables →

More Two-Way Tables Examples

Example 2 hard

From a 2×2 table: Group/Outcome frequencies: A-Success=40, A-Fail=10, B-Success=25, B-Fail=25. Test

Example 3 easy

A two-way table: Left-handed/Male=12, Left-handed/Female=8, Right-handed/Male=88, Right-handed/Femal

Example 4 hard

Construct a two-way table from this information: 200 students surveyed; 120 prefer online learning;

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