SAT Math Review · Problem-Solving and Data Analysis
Probability from two-way tables
Every probability on this page is one number from the table divided by another. The whole skill is choosing the two.
The table summarizes the 200 students who responded to a school survey, by grade level and whether the student has a part-time job.
Has a job
No job
Total
Junior
45
75
120
Senior
50
30
80
Total
95
105
200
If one of these students is selected at random, what is the probability that the student is a senior or has a part-time job?
Count the seniors, add the students with a job, then remove the 50 students who are both: 80 + 95 − 50 = 125, so the probability is 125200 = 58. D adds 80 and 95 and stops, counting every working senior twice and giving 175200 = 78 — the single most common slip on "or" items. A is 50200, the overlap cell on its own, which answers "senior and has a job". B is 95200, the job column only, which forgets the seniors who do not work.
1
A two-way table sorts one group of people two ways at once.
Rows are one category, columns are the other, and the four inner cells are the four kinds of student. The last row and the last column are the margins: they are not new people, only the ones already in the table, counted along one direction.
Has a job
No job
Total
Junior
45
75
120
Senior
50
30
80
Total
95
105
200
cell = both labels at oncerow margin = that row's peoplecolumn margin = that column's peoplecorner = everybody
45 + 75 = 120
the two cells of the Junior row
45 + 50 = 95
the two cells of the "Has a job" column
120 + 80 = 95 + 105 = 200
the grand total, reached two ways
The grand total in the bottom-right corner is the number of people in the whole study — 200 students, each counted in exactly one of the four inner cells.
2
Probability is one count over another count.
When a student is selected at random from the whole table, the denominator is the grand total. Every student is equally likely, so the probability is just the share of the 200.
P =
favourabletotal
P(senior) = 80200 = 25
a row margin over the grand total
P(has a job) = 95200 = 1940
a column margin over the grand total
P(junior with no job) = 75200 = 38
a single cell over the grand total
The SAT's favourite phrasing
"If one of these students is selected at random" is the sentence that puts 200 underneath. The choices are usually left unsimplified as well as simplified, so 95200 and 1940 may both appear — they are the same answer, and the test will only print one of them.
3
"And" is a single cell — no arithmetic needed.
Two labels joined by "and" pick out the one cell where the row and the column cross. The table has already done the counting.
senior and has a job → row Senior, column Has a job
they meet at 50
P = 50200 = 14
still out of all 200
"A senior who has a part-time job", "a senior with a job" and "both a senior and employed" are three ways of asking for the same cell. Read the two labels, find the crossing, stop.
Do not multiply
P(senior) × P(has a job) = (25)(1940) = 19100 = 0.19, but the true answer is 0.25. Multiplying only works for independent events, and here the two categories are related — seniors are far more likely to work. With a table in front of you, never multiply: read the cell.
4
"Or" adds a whole row to a whole column — so the crossing gets counted twice.
Shade the Senior row and shade the "Has a job" column. One cell gets shaded twice, and that is the cell you must give back.
P(A or B) = P(A) + P(B) − P(A and B)
the rule, in counts or in probabilities
80 + 95 − 50 = 125
seniors, plus workers, minus the working seniors
P = 125200 = 58
divide by 25 twice, or type 125 ÷ 200
P(senior or has a job) = 58 = 0.625
The 78 trap from the opening problem
80 + 95 = 175 gives 175200 = 78, and it is wrong because the 50 working seniors were counted as 100 people. A two-way table makes the check easy: the three shaded cells hold 45 + 50 + 30 = 125 real students, and no student can be in two cells.
5
Nothing to subtract when the two groups cannot overlap.
If no student can satisfy both labels, the shared cell is empty and "or" is a plain addition of cells.
senior or has a job
80 + 95 − 50
they overlap in 50 students
working junior or non-working senior
45 + 30
two separate cells, no overlap
P =
45 + 30200
=
75200
=
38
Two different cells never share a student, so any list of cells can simply be added. The overlap only appears when you add a margin to a margin — a whole row plus a whole column.
6
A missing entry is one subtraction away.
The SAT often prints a table with holes in it, or with a letter in one cell, and the margins do the rest of the work.
Has a job
No job
Total
Junior
45
?
120
Senior
?
?
?
Total
95
?
200
120 − 45 = 75
juniors with no job, from the Junior row
200 − 120 = 80
seniors, from the Total column
95 − 45 = 50
seniors with a job, from the job column
80 − 50 = 30
the last corner, and 75 + 30 = 105 checks it
When the missing entry is a letter
If the senior-with-a-job cell were written as x and the stem said the probability of selecting a senior with a job is 14, then x200 = 14 and x = 50. Set the given probability equal to the count over the grand total and solve the one equation.
7
Only two words can move the denominator off the grand total.
"Given that" and "of those who" restrict the selection to one row or one column. Without them, 200 stays underneath.
The question
Over
Value
a junior with a job
200, the grand total
45200 = 940
of the juniors, one with a job
120, the Junior row
45120 = 38
of those with a job, one who is a junior
95, the job column
4595 = 919
Same 45 on top all three times. Only the denominator moved, and it moved because of a phrase, not because of a number. The next lesson is entirely about that phrase; here, just notice whether it is present.
Where the calculator earns its keep
These fractions rarely simplify neatly, so convert: 45 ÷ 95 = 0.4737. If the choices are decimals, or the question says "closest to", the division is faster than hunting for a common factor. Typing 4595 into Desmos returns the decimal 0.4736842105, not a simplified fraction, so match that against the choices rather than waiting for 919 to appear.
✓
Check: does the table still add up, and is the size sensible?
rows120 + 80 = 200✓
columns95 + 105 = 200✓
"or" is bigger0.625 > 0.475 and 0.40bigger than each piece ✓
"and" is smaller0.25 < 0.475 and 0.40smaller than each piece ✓
Remember the flow
Find the cell · Divide by the grand total · Subtract the overlap once
No conditioning phrase means the denominator is everybody.
Now you try
Practice problem
Build the table first
A workshop was attended by 300 people. Of these, 180 were first-time attendees and the rest were returning attendees. A total of 105 of the attendees signed up online, including 60 of the first-time attendees. If one of the attendees is selected at random, what is the probability that the attendee is a returning attendee who did not sign up online?
Fill the margins: 300 − 180 = 120 returning, and 105 − 60 = 45 of them signed up online, so 120 − 45 = 75 returning attendees did not. The probability is 75300 = 14. A is 45300, the returning attendees who did sign up online — the neighbouring cell. C is 120300, the first-time attendees who did not sign up online, the right column read in the wrong row. D is 195300, everyone who did not sign up online, first-time attendees included.
Hint: draw the empty 2 × 2 table with its margins, write in the four numbers the stem gives you, and let each margin produce the next cell. You are looking for one inner cell over 300.
