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Probability and conditional probability: SAT Math notes
These questions test whether you can find the probability of an event from counts or tables, including conditional probability, where you only look at one group, and work backward from a probability to a count.
Study it with worked examples and 8 practice questionsPractise Probability and conditional probability questions
What this topic tests
Probability is a fraction: the number of outcomes you want divided by the number of possible outcomes, when every outcome is equally likely. It's always between 0 and 1, and you can write it as a fraction or a decimal.
Most questions use a two-way table, which sorts the same people or things in two ways at once, for example by age and by number of visits. The key skill is choosing the right denominator: the whole table, one row, or one column.
Conditional probability questions narrow the group with words like given that, of those, or if the person chosen works in. As soon as you see those words, the group you're picking from shrinks, and so does the denominator.
Key ideas
- Probability = number of outcomes you want ÷ total number of possible outcomes.
- The probabilities of all the separate outcomes add up to 1. So the probability something does NOT happen is 1 minus the probability that it does.
- In a two-way table, a single cell counts things that fit both labels. A row or column total counts everything with one label.
- Given that, of those, among, and if the person chosen is all mean the same thing: shrink the denominator to that group only.
- Number of outcomes = probability × total. If 0.35 of 60 marbles are blue, then 21 are blue.
- A table of percents works just like a table of counts. Treat the percents as counts out of 100.
The 4 kinds of Probability and conditional probability question
Sorted from every Probability and conditional probability question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Basic probability from counts
How to spot it. One random pick from a single collection: a marble from a bag, a rock from a collection, a singer from a list. You're told how many of each kind there are, maybe in a simple one-column frequency table.
How to answer it:
- Find the total number of possible outcomes. Add up the groups if the total isn't given.
- Find how many outcomes match what the question asks for.
- Divide: favorable ÷ total. Simplify, or change to a decimal if the choices are decimals.
The trap. Using another group as the denominator instead of the total. With 13 red pens and 37 blue pens, the probability of red is 13/50, not 13/37. The whole collection always goes on the bottom.
2. Two-way tables: out of everyone
How to spot it. A two-way table, with one item picked from the whole group and nothing narrowing it down. The question asks for one cell (a hybrid car priced at no more than 25,000 dollars), one row or column total (a customer who did not buy gasoline), or a missing number in the table.
How to answer it:
- Find the grand total of the table. If there's no total row or column, add up all the cells.
- Find the count that matches the description. Two labels joined by and or but point to a single cell; one label points to a row or column total.
- Divide that count by the grand total.
The trap. Using a row or column total as the denominator when nothing in the question narrows the group. If it just says one of the cars is selected at random, with no given that, the denominator is all of the cars.
3. Conditional probability: given a group
How to spot it. The pick is narrowed to one group first, with words like given that, of those, among, or if the person chosen works in. Usually there's a two-way table, but it can also be a frequency table or a table of percents.
How to answer it:
- Find the group named after given that (or of those). That group is your denominator: add up its row, column or cells.
- Inside that group only, count the ones that match what's asked. That's your numerator.
- Divide the numerator by the denominator.
- Sense check: everything in the numerator must also belong to the denominator group.
The trap. Using the whole table as the denominator. If the question says of the days with no rain, divide by the 64 no-rain days, not by all 84 days. Also don't flip the condition: thin crust given pepperoni is a different probability from pepperoni given thin crust.
4. Work backward from a probability
How to spot it. The probability is given and a count is missing: how many marbles are blue, how many attendees are in group C, or the value of x in a table.
How to answer it:
- Write the probability as count ÷ total.
- If you know the total, multiply: count = probability × total.
- If the unknown is part of the total, set up an equation such as 2/(10+x)=1/9 and solve it.
- If the groups cover everyone, use the fact that their probabilities add to 1 to find the missing one.
The trap. Forgetting that the unknown can be in the denominator too. In the blood type question, x is a group of rhesus-negative people, so it changes the total as well as one cell of the table.
Practise Probability and conditional probability
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.
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