Integrated Math 2 / Unit 9: Probability
9.6 Two-Way Tables
Notes
A two-way table counts people by two categories. Totals give marginal probabilities, and rows or columns give conditional probabilities.
- Example: 70 students. Band members: 12 play sports and 8 do not. Non-members: 28 play sports and 22 do not. So 40 play sports and 20 are in band.
- $\displaystyle P(\text{sports} \mid \text{band}) = \frac{12}{20} = 0.6$ (use the band row total).
- $\displaystyle P(\text{band} \mid \text{sports}) = \frac{12}{40} = 0.3$ (use the sports column total).
- The two conditionals are different. Check independence by comparing $P(A \mid B)$ with $P(A)$.
Practice
Of 70 students, 20 are in band. Of the band members, 12 play a sport. What is the probability a band member plays a sport?
Use the band total as the denominator: $\displaystyle \frac{12}{20} = \frac{3}{5}$.
Of 70 students, 40 play a sport and 12 of those students are in band. What is the probability a student who plays a sport is in band?
Use the sports total as the denominator: $\displaystyle \frac{12}{40} = \frac{3}{10}$.
Of 70 students, 40 play a sport and 20 are in band. Of the band members, 12 play a sport. Are playing a sport and being in band independent?
Independent events need $P(A \mid B) = P(A)$, and $0.6 \neq 0.571$.
Ready for more? Take the Unit 9 test.