"Given that" is not extra information. It is an instruction to change the denominator.
The table summarizes the 240 students at a school who took a practice test, by whether the student had taken the school's prep course and whether the student scored above 1200.
| Above 1200 | 1200 or below | Total | |
|---|---|---|---|
| Took the course | 60 | 20 | 80 |
| No course | 40 | 120 | 160 |
| Total | 100 | 140 | 240 |
Of the students who scored above 1200, what fraction had taken the prep course?
"Of the students who scored above 1200" throws away the other 140 students, so the denominator is the 100 in that column, and 60 of them took the course: 60100 = 35. D is 6080, the same 60 divided by the course row — that is the probability of scoring above 1200 given the course, the reversed question, and it is the trap this lesson exists for. A is 60240, which keeps the grand total and answers "above 1200 and took the course". B is 100240, the probability of scoring above 1200 at all.
You are not being asked how likely it was that a high scorer was drawn. You are told one was — so the 140 students who scored 1200 or below stop being possibilities.
Every conditional probability question is really a question about which number goes underneath. The numerator is usually the easy part.
Cover the rest of the table with your hand. Whatever is left is the whole world for this question.
| Above 1200 | 1200 or below | Total | |
|---|---|---|---|
| Took the course | 60 | 20 | 80 |
| No course | 40 | 120 | 160 |
| Total | 100 | 140 | 240 |
"given that the student took the course"
use the Took the course row only
course students altogether = 80
the new denominator, from the row margin
of those, above 1200 = 60
the numerator, from inside the same row
P(above 1200 | course) = 6080 = 34
0.75, and the 240 never appears
The grand total is not wrong, it is simply irrelevant: both the 60 and the 80 are already inside the same row, so dividing by 240 twice and cancelling would give the same answer the long way round.
The SAT rarely writes a vertical bar. It writes a phrase, and the group named in that phrase becomes the denominator.
| The stem says | Denominator | Answer |
|---|---|---|
| one of these students is selected at random, and scored above 1200 and took the course | 240, everybody | 60240 = 14 |
| given that the student took the course | 80, the row | 6080 = 34 |
| of the students who scored above 1200 | 100, the column | 60100 = 35 |
| among the students who did not take the course | 160, the other row | 40160 = 14 |
"A student who scored above 1200 and took the course" describes one cell out of all 240 — nobody has left the room, so the denominator stays at the grand total. Only a phrase that tells you something is already true moves it.
P(A | B) = P(A and B)P(B) so P(A and B) = P(A | B) · P(B)
A community center reports that 40% of its members are over 50, and that 12% of all its members are over 50 and volunteer.
P(volunteer | over 50) = 0.120.40
the "and" percent over the condition's percent
= 0.30
30% of the over-50 members volunteer
Read the other way, a stem that says "45% of the respondents own a tablet, and of the respondents who own a tablet, 60% own a smart speaker" has handed you a conditional probability already:
P(tablet and speaker) = 0.45 × 0.60
condition first, then the share inside it
= 0.27
27% of all respondents own both
"60% of the respondents who own a tablet" is a percent of a smaller group, so it cannot be added to or compared with a percent of everybody. Multiplying by 0.45 converts it back to a share of the whole survey — 100 respondents, 45 with tablets, 27 of those with a speaker as well.
Both of these use the same 60 students. Only the group they are being compared against is different.
60 ÷ 80 = 0.75
out of the course students
60 ÷ 100 = 0.60
out of the high scorers
The condition is the thing you are told is already true, and it goes underneath. "Of the students who scored above 1200, what fraction took the course" conditions on the score, not on the course. Both 34 and 35 will be printed in the choices, so choosing correctly is worth more than the arithmetic.
Compare the conditional probability with the plain one. If they match, the two categories are independent.
Knowing about the course changes the estimate, so the two rows behave differently and the shortcut P(A and B) = P(A) · P(B) is not available here. Note what this does not say: the students chose the course themselves, so the table shows an association and not a cause.
Three totals are offered; only one is the denominator
A school has 900 students. Of these students, 360 play a sport and 300 are in the honors program, and 216 of the students who play a sport are in the honors program. If a student in the honors program is selected at random, what is the probability that the student plays a sport?
The student is already known to be in the honors program, so the denominator is the 300 honors students, and 216 of them play a sport: 216300 = 1825 = 0.72. C is 216360, the reversed condition — the probability that a student who plays a sport is in honors. A is 216900, which keeps the whole school underneath and answers "plays a sport and is in honors". B is 300900, the probability of being in honors at all, which ignores the sport entirely.
Hint: the stem prints three totals — 900, 360 and 300 — precisely so that all three appear as denominators in the choices. Underline the group you are told the student belongs to, and use that one.
