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Math foundations
SAT probability: let the word “given” shrink the table
Most SAT probability questions come from a two-way table, and most mistakes come from dividing by the wrong total. Click through an interactive table and see exactly which number belongs on the bottom.
SPSAT Practice Editorial7 min read
The short version
What you'll take away
- Probability = favorable outcomes ÷ total outcomes in the group you are choosing from.
- “Given” or “among” narrows the group, so the denominator becomes a row or column total, not the grand total.
- P(A given B) and P(B given A) usually differ because they use different denominators.
In this article
One table, many questions
A two-way table sorts the same people (or objects) by two features at once. In the example below, 200 students were surveyed about grade level and whether they play a school sport. Every cell counts the students who fit both its row and its column.
The numerator is the count of what you are looking for. The denominator is the size of the group you are choosing from. The word “given” or “among” changes the denominator. Pick an event and a condition below, and watch the highlighted group change.
Try it yourself · interactive
Pick an event, then pick who you are choosing from
| Juniors | Seniors | Total | |
|---|---|---|---|
| Plays a sport | 54 | 30 | 84 |
| No sport | 66 | 50 | 116 |
| Total | 120 | 80 | 200 |
Plays a sport among Juniors
54 ÷ 120 ≈ 0.450
Without a condition, use the grand total
If a student is chosen at random from all 200, the probability that the student plays a sport is \(\displaystyle \frac{84}{200} = 0.42\). The 84 is the “Plays a sport” row total: 54 juniors plus 30 seniors.
“Given” shrinks the group
Now choose only from juniors. The probability that a junior plays a sport is \(\displaystyle \frac{54}{120} = 0.45\). Only the 120 juniors are in the group, so the 80 seniors drop out of the denominator.
Phrases that signal a narrower group include “given that,” “among,” “of those who,” and “if a student is a junior.” When you see one, put your finger on that row or column total before you calculate anything.
Common traps
Wrong answer choices on table questions are usually correct calculations with the wrong denominator. Expect to see the grand total, the row total, and the column total offered side by side.
Your turn
Quick practice
Using the table above, a student who plays a sport is chosen at random. What is the probability that the student is a senior?
Hint: “A student who plays a sport” names the group you are choosing from.
- Dividing by 200 when the question says “among seniors.”
- Swapping the condition and the event (juniors who play versus players who are juniors).
- Reading a total row or column as if it were a single cell.
Practice: try these before checking the answers
Use the same table. Name the group first, then count.
- 1. What is the probability that a randomly chosen student is a senior?
- 2. What is the probability that a senior does not play a sport?
- 3. What is the probability that a student who does not play a sport is a junior?
- 4. A bag holds 3 red, 5 blue, and 2 green marbles. What is the probability that a marble drawn at random is not blue?
Sources & further reading
One step from here
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