Turn the Wording into Table Cells
A two-way table organizes outcomes by two variables. When a question asks about two categories from different variables, the wording tells you which cells to use. “And” asks for outcomes that meet both conditions, so it points to the overlap. Inclusive “or” asks for outcomes that meet either condition or both, so it includes the overlap along with the cells belonging to only one event.
As in Using Two-Way Tables to Find Probabilities, an interior cell represents outcomes in both its row category and its column category. A row or column total represents one category without specifying the other variable. For a randomly selected individual from the full group in a count table, divide the selected count by the grand total. In a probability table, the cells already give probabilities, so add the selected cell probabilities directly.
A reliable first move is to translate each phrase into a condition on the row or column variable. Then identify the cells meeting the first condition, the cells meeting the second, and any shared cells. For “and,” keep only the shared cells. For “or,” include all selected cells but do not count any cell twice.
Worked Example: Robotics or Art
Worked Example: Robotics or Art
A school club survey records whether each of 120 students participates in robotics and whether each participates in art. The table gives the number of students in each combination. Suppose one student is selected at random from all 120. Find the probability that the student participates in robotics or art, and compare it with the probability that the student participates in both.
| Art: Yes | Art: No | Total | |
|---|---|---|---|
| Robotics: Yes | 16 | 18 | 34 |
| Robotics: No | 32 | 54 | 86 |
| Total | 48 | 72 | 120 |
State: Let \(R\) be the event that the selected student participates in robotics, and let \(A\) be the event that the student participates in art. The first question asks for \(P(R\cup A)\); the comparison asks for \(P(R\cap A)\).
Plan: The word “or” includes students in robotics only, art only, and both clubs. The word “both” asks only for the shared interior cell. Because the student is selected from the full group, use 120 as the denominator for a count-based probability.
Do: The intersection is the cell in the “Robotics: Yes” row and “Art: Yes” column: 16 students. Therefore, \(P(R\cap A)=16/120\). For the union, select the three interior cells that show robotics only, art only, or both: 18, 32, and 16 students. Their total is \(18+32+16=66\).
A check using the margins and overlap gives the same union count: the robotics total plus the art total, minus the shared students, is \(34+48-16=66\). This check confirms that the 16 students in both clubs were not counted twice. The intersection probability is \(16/120\approx0.1333\), rounded to four decimal places.
Conclude: The probability that a randomly selected student participates in robotics or art is 0.55. The probability that the student participates in both is about 0.1333. The “or” answer is larger because it includes students in either club, not only students in both.
Mark the Cells Before Doing Arithmetic
One useful technique is to mark the table before calculating. For an “and” question, mark only the cell or cells where both stated categories meet. For an “or” question, mark the cells belonging to either event. If the event is defined by a row category and a column category, the union will commonly consist of three interior cells: the shared cell and one cell belonging exclusively to each event.
Do not mistake the marginal totals for separate interior cells. A row total or column total already includes more than one interior cell. Adding both margins for an “or” question counts the shared cell twice, so subtract it once—or add the three selected interior cells directly. The direct cell method is particularly helpful when the question uses a complement, such as “bike or does not carry a bottle,” because it makes clear which table regions are included.
Worked Example: Bike or Reusable Bottle
Worked Example: Bike or Reusable Bottle
A community center asks 200 visitors whether they arrived by bicycle and whether they brought a reusable water bottle. The table shows the results. One visitor is selected at random from all 200. Find the probability that the visitor arrived by bicycle or brought a reusable bottle, and find the probability that both are true.
| Bottle: Yes | Bottle: No | Total | |
|---|---|---|---|
| Bike: Yes | 38 | 34 | 72 |
| Bike: No | 52 | 76 | 128 |
| Total | 90 | 110 | 200 |
State: Let \(B\) be the event that a visitor arrived by bicycle and \(W\) the event that the visitor brought a reusable bottle. We want \(P(B\cup W)\) and \(P(B\cap W)\).
Plan: For “and,” use the cell where “Bike: Yes” and “Bottle: Yes” meet. For inclusive “or,” include the three cells where at least one of those conditions is true. Use 200 as the denominator because the selection is from all visitors in the table.
Do: The shared cell contains 38 visitors, so \(P(B\cap W)=38/200=0.19\). The union uses the shared cell, the bike-only cell, and the bottle-only cell. Those counts are 38, 34, and 52, for a total of \(38+34+52=124\).
Check by using the row and column totals: \(72+90-38=124\). The same count results because the 38 visitors who meet both conditions are first included in both margins, then removed once to avoid counting them twice. Also, the union count of 124 is at least as large as either individual event count, 72 or 90, as it should be.
Conclude: The probability that a randomly selected visitor arrived by bicycle or brought a reusable bottle is 0.62. The probability that the visitor did both is 0.19.
Read Probabilities Directly from a Probability Table
A table may show probabilities rather than counts. The cell-selection process does not change: identify the event from the wording, then select the relevant cells. The difference is that you add the probabilities in those cells instead of dividing a count by the grand total. As explained in Probabilities Must Sum to One, all entries in a complete probability table should sum to 1.
Worked Example: Compost or Drip Irrigation
A probability model describes a randomly selected community garden plot. The two variables are whether the plot uses compost and whether it uses drip irrigation. The table entries are probabilities. Find the probability that a plot uses compost or drip irrigation, and the probability that it uses both.
| Drip: Yes | Drip: No | Total | |
|---|---|---|---|
| Compost: Yes | 0.22 | 0.18 | 0.40 |
| Compost: No | 0.27 | 0.33 | 0.60 |
| Total | 0.49 | 0.51 | 1.00 |
State: Let \(C\) be the event that a plot uses compost and \(D\) the event that it uses drip irrigation. The questions ask for \(P(C\cup D)\) and \(P(C\cap D)\).
Plan: The intersection is the cell where both categories are “Yes.” The union includes that cell and the two cells where exactly one of the practices is used. Since the table entries are probabilities, add the selected probabilities directly.
Do: The “both” cell gives \(P(C\cap D)=0.22\). The “compost yes, drip no” cell contributes 0.18, and the “compost no, drip yes” cell contributes 0.27. Thus the union probability is \(0.22+0.18+0.27=0.67\).
The row and column totals provide a second check: \(P(C)=0.40\) and \(P(D)=0.49\), so \(P(C\cup D)=0.40+0.49-0.22=0.67\). The four interior probabilities also sum to \(0.22+0.18+0.27+0.33=1.00\), consistent with a complete probability table.
Conclude: According to this probability model, the probability that a randomly selected plot uses compost or drip irrigation is 0.67, while the probability that it uses both is 0.22.
When the Wording Includes “Not”
A question can combine “or” with a category’s complement. For example, “bike or no bottle” does not mean “bike and bottle,” and it does not mean “bike only.” Translate each part separately: “bike” selects the “Bike: Yes” row, and “no bottle” selects the “Bottle: No” column. The union includes every cell in either selected region, including the cell where those regions overlap.
Worked Example: Bike or No Bottle
Use the visitor table in the previous example. Find the probability that a visitor arrived by bicycle or did not bring a reusable bottle.
The “Bike: Yes” row contains 38 and 34 visitors. The “Bottle: No” column contains 34 and 76 visitors. The cell where these two conditions overlap is the bike-and-no-bottle cell, with 34 visitors. Select the union’s three distinct cells: 38, 34, and 76. Do not add the shared 34 twice.
The margin-and-overlap check agrees: the bike total is 72, the no-bottle total is 110, and their shared count is 34. Therefore, \((72+110-34)/200=148/200=0.74\). The probability is 0.74 that a randomly selected visitor arrived by bicycle or did not bring a reusable bottle.
A Cell-Selection Routine
Before calculating, say the event in plain language. Then locate the relevant categories in the table. This small pause helps prevent a familiar error: choosing the cell for “and” when the question asks for “or,” or choosing a full margin when the question asks for only an intersection.
Identify the row or column category represented by each part of the question.
“And” means both conditions hold. Inclusive “or” means either condition or both conditions hold.
For “and,” select the shared cell or cells. For “or,” select all cells satisfying at least one condition, including the overlap once.
Add selected counts and divide by the grand total, or add selected probabilities. Check the union with the margins minus the overlap when those totals are available.
Common Mistakes and AP Exam Tips
- Treating “or” as “exactly one.” In probability questions, “A or B” ordinarily means inclusive or: A, B, or both. The shared cell belongs in the union.
- Using only the shared cell for an “or” question. The shared cell answers “A and B.” An “or” question also includes the cells for A only and B only.
- Adding both margins without adjusting for overlap. Each margin includes the shared cell. Add the margins and subtract the intersection once, or add the three distinct interior cells directly.
- Counting the overlap twice when listing cells. A cell can satisfy both conditions, but it is still one cell. Mark it once in the union.
- Dividing a probability table entry by the grand total. The entries already are probabilities. Add the relevant entries directly; dividing again would change the answer incorrectly.
- Using a row total for an intersection. A row total includes both column categories in that row. For “and,” go to the cell where the row and column conditions meet.
- Leaving out what the result means. A complete response states the probability in context, such as “The probability that a randomly selected visitor arrived by bicycle or brought a reusable bottle is 0.62.”
A quick reasonableness check is to compare a union probability with its individual event probabilities: the union cannot be smaller than either event alone. For a count table, the union count cannot exceed the grand total. These checks can reveal a cell-selection or arithmetic mistake, although they do not replace explaining which cells represent the event.
Check Your Understanding
For each question, identify the cells that belong to the event before calculating. Unless a question says otherwise, select one individual at random from the full group represented.
- In the robotics-and-art table, how many students participate in both activities? What probability does this count represent?
- Using that same table, find the probability that a student participates in robotics or art. Show which three interior cells you use.
- In the visitor table, find the probability that a visitor arrived by bicycle and did not bring a bottle. Which single cell answers this “and” question?
- In the garden probability table, find the probability that a plot uses neither compost nor drip irrigation. Which cell gives the answer?
- Explain why adding the bicycle total and bottle total without subtracting their overlap does not give the correct count for “bike or bottle.”