On the test
Problem-Solving and Data Analysis is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I think about probability, I think about picking a mango from a basket at the market. If there are 10 mangoes and 3 are not ripe, your chance of grabbing an unripe one is 3 out of 10. Easy. But now your friend says, 'I only want to look at the big mangoes.' Suddenly you ignore the small ones and count only the big ones. Your basket changed, so your bottom number changed. That is the whole idea of conditional probability. I always ask myself, which group am I really picking from? On the SAT, a two-way table is just a basket sorted into boxes. When the question says 'given' or 'of those who', I shrink the basket to that row or column first, and then I count.
Probability questions look scary because of the words, but they are only counting. If you can count a row and divide, you can win these. I believe you can.
What is inside this part
The chance of one event, found as wanted outcomes divided by all outcomes.
Choosing from a bag or a group · Fractions, decimals and percents
Reading a table that sorts a group by two features and using its totals to find a chance.
Row and column totals · Finding a missing number in a table
The chance of one thing happening when you already know another thing is true, so you only use that smaller group.
Words like given that and of those who · P(A | B) = P(A and B) / P(B)
The chance of two or more events happening together or in a row.
With and without replacement · The rule P(A or B) = P(A) + P(B) - P(A and B)
How to know a question is from this part
A table of counts or a bag of objects, and a question that starts 'What is the probability...'.
- The words 'chosen at random' or 'randomly selected'
- A table with Total row and Total column
- 'Given that', 'of those who', 'among the', 'a student who ... is selected'
- Notation like P(A | B) or P(A and B)
- 'Without replacement' or 'is not put back'
The lesson
1. What probability means
Probability is a number that tells you how likely something is. It is a fraction: the number of outcomes you want, divided by the number of outcomes that are possible.
So probability = wanted outcomes / all possible outcomes. It is always from 0 to 1. A probability of 0 means it cannot happen. A probability of 1 means it will surely happen. 1/2 means equally likely to happen or not.
The SAT may write the answer as a fraction, a decimal or a percent. They are the same thing: 1/4 = 0.25 = 25%.
A bag has 4 red, 5 yellow and 1 green ball. One ball is chosen at random. What is the probability it is yellow?
Show the steps
- Wanted outcomes: yellow balls = 5.
- All possible outcomes: 4 + 5 + 1 = 10.
- Probability = 5/10 = 1/2.
1/2
2. Reading a two-way table
A two-way table sorts people (or things) by two features at the same time. For example: boys or girls, and football or music. Each box in the middle tells how many are in BOTH groups.
The numbers on the edge are totals. A row total adds across. A column total adds down. The number in the bottom-right corner is the grand total of everybody.
Before doing anything, I check that the totals make sense: the row totals should add to the grand total, and so should the column totals.
Football | Music | Total Boys 24 12 36 Girls 16 28 44 Total 40 40 80 How many girls chose music?
Show the steps
- Find the Girls row.
- Move across to the Music column.
- The box where they meet says 28.
28
3. Simple probability from a table
When the question picks someone from the whole group, the bottom number is the grand total. The top number is whatever group the question names.
Example words: 'a student is chosen at random. What is the probability the student is a girl?' Top: all girls (the row total 44). Bottom: everyone (80). So 44/80 = 11/20.
Using the same table, what is the probability that a randomly chosen student chose football?
Show the steps
- Football column total = 40.
- Everyone = 80.
- 40/80 = 1/2.
1/2
4. Conditional probability: shrink the basket
Conditional probability means 'the probability of something, given that we already know something else.' The SAT uses words like 'given that', 'of those who', 'among', or 'a student who chose music is selected'.
The method: the condition tells you the new group. Use ONLY that row or column. The bottom number is the total of that row or column. The top number is the box inside it that matches what you want.
In symbols, P(A | B) = P(A and B) / P(B). The line | is read 'given'. Whatever comes after the line is the group you shrink to.
Primary | Secondary | Total Walk 30 20 50 Bus 25 45 70 Total 55 65 120 A student who travels by bus is chosen. What is the probability the student is in secondary?
Show the steps
- 'Who travels by bus' means use only the Bus row. Bottom number = 70.
- Bus and secondary box = 45.
- 45/70 = 9/14.
9/14
5. More than one pick, and the 'and' and 'or' rules
If you pick twice and put the first one back (with replacement), the second pick has the same chances. Multiply the two probabilities.
If you do NOT put it back (without replacement), the bag has changed: one fewer item in the top (if it was a wanted one) and one fewer in the bottom. Multiply, but update the second fraction.
For 'A or B': P(A or B) = P(A) + P(B) - P(A and B). We subtract the overlap because it was counted twice. For independent events (one does not affect the other), P(A and B) = P(A) × P(B).
A bag has 12 marbles, 4 red. Two are taken without putting the first back. What is P(both red)?
Show the steps
- First: 4/12.
- Second: 3 red left out of 11 marbles = 3/11.
- 4/12 × 3/11 = 12/132 = 1/11.
1/11
6. Working with missing numbers
Sometimes the table has an unknown, like x. The question gives a probability that lets you write an equation. Write the probability as a fraction using x, set it equal to the given number, and cross multiply.
Then fill the table, find the new totals, and answer the real question. Always re-read the last sentence, because the question often asks for something different from what you solved.
Art: Girls x, Boys 50. A student who chose Art is a girl with probability 3/5. Find x.
Show the steps
- Girls among Art = x. All Art = x + 50.
- x/(x + 50) = 3/5.
- 5x = 3x + 150, so 2x = 150 and x = 75.
x = 75
7. Common traps
Trap 1: Using the grand total when the question has a condition. If you see 'given' or 'of those who', do not use the grand total.
Trap 2: Flipping the condition. P(A | B) and P(B | A) are different. The thing after 'given' or 'who' is your bottom number.
Trap 3: Forgetting that the bag changes when nothing is put back.
Trap 4: Answering with a count when the question wants a probability, or the opposite. Read the last line.
See it
- 1Find the words 'given', 'of those who', 'among', or 'who'.
- 2The group after these words is my new basket. Use only that row or column.
- 3Bottom number = total of that row or column.
- 4Top number = the box inside that row or column that matches what I want.
- 5Divide, then simplify and check against the options.
| Question | Top | Bottom | Answer |
|---|---|---|---|
| P(girl) | 44 (girls row) | 80 (everyone) | 11/20 |
| P(girl | chose music) | 28 (girls in music) | 40 (music column) | 7/10 |
| P(chose music | girl) | 28 (girls in music) | 44 (girls row) | 7/11 |
- Put the item back after the first pick.
- The bag is the same for the second pick.
- Multiply the same-looking fractions.
- Example: 4/12 × 4/12
- The first item stays out.
- The second pick has one fewer item on the bottom.
- If you removed a wanted item, the top also goes down by one.
- Example: 4/12 × 3/11
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