Choose a representation
Subtract overlap when adding event counts. For exactly one success in two independent trials, add the two disjoint orders. Use conditional branch probabilities to reconstruct a joint probability.
Worked example: follow the quantity
In a group, 8 students study art, 11 study music, and 3 study both. How many study at least one of these subjects? Add the two subject counts. Subtract the overlap once because it was included twice. Substitute the given numbers: (8)+(11)-3 = 16. Therefore the requested value is 16.
Contrast a plausible mistake
For that example, the tempting answer 22 follows this error: Add the overlap a third time. The requested quantity is 16. Explain why the wrong operation changes the mathematical object, not merely the arithmetic.
A second representation
A bag is chosen: A with probability 1/3 or B with probability 2/3. A produces red with probability 1/3; B produces red with probability 2/3. What is P(A given red)? Find joint probabilities for A and red, and for B and red. Divide the first joint probability by the sum of both red paths. P(A and red)=1/9; P(B and red)=4/9; their ratio within red is (1/9) ÷ (5/9) = 1/5. Therefore the requested value is 1/5.
Transfer before checking
In a group, 9 students study art, 12 study music, and 3 study both. How many study at least one of these subjects? Try without viewing the answer. Check: 18. Add the two subject counts. Subtract the overlap once because it was included twice. Substitute the given numbers: (9)+(12)-3 = 18.
Scope and conditions
Independent events can overlap; mutually exclusive positive-probability events cannot be independent. Conditioning reverses neither the numerator nor the denominator automatically. These original practice tasks sample the official topic; they do not certify complete coverage or a predicted ACT score.
In a group, 8 students study art, 11 study music, and 3 study both. How many study at least one of these subjects?
- Add the two subject counts.
- Subtract the overlap once because it was included twice.
- Substitute the given numbers: (8)+(11)-3 = 16.
- Therefore the requested value is 16.
Compare the other answer choices
Report the overlap rather than the union.
Subtract the overlap twice, obtaining exactly one subject.
Add the overlap a third time.
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.
ACT Mathematics content