Choose a method
Build a small count table or name the allowed group. Add disjoint event counts; use the complement when a not-event is easier to count.
Avoid this trap
The denominator after conditioning is the conditioned group, not the entire survey. A probability and a predicted number of people use different units.
Check and transfer
Check that probabilities lie from zero to one. An expected count based on a sample proportion is a model estimate, not a guaranteed outcome.
Worked example with a numerical check
A survey has 16 responses, of which 4 report walking to school. What fraction of respondents report a different travel mode? Subtract the walking count from the total. Divide the remaining count by all responses. The requested value is 3/4.
Contrast and diagnose
Of 6 students in the chess club, 4 also play music. Given that a selected student is in the chess club, what is the probability the student plays music? The condition restricts the sample space to chess-club members. Use members who also play music as the favorable count. The requested value is 2/3. An incorrect approach: Omit one music-playing member from the favorable group.
Transfer practice
A bag contains 4 red, 6 blue and 2 green tokens. One token is selected at random. What is the probability it is red or green? Try first. Check: 1/2. Red and green are disjoint colors, so add their counts. Divide by the total number of tokens.
A survey has 16 responses, of which 4 report walking to school. What fraction of respondents report a different travel mode?
- Subtract the walking count from the total.
- Divide the remaining count by all responses.
- The requested value is 3/4.
Compare the other answer choices
Assume two named categories are equally frequent.
Compare walkers with nonwalkers instead of using all responses.
Add the walking group to the total denominator again.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications