How the SAT tests this, with worked examples — plus the 2 skills it covers. Free to read, no account needed.
Probability on the SAT is counting, not theory. Almost every question is
P=total number of outcomesnumber of outcomes you want
and the difficulty lies entirely in deciding which numbers those are.
The data nearly always arrives as a two-way table, and the trick is reading which part of the table the question restricts you to.
If a question asks for the probability that a randomly chosen student takes the bus, the denominator is everyone: 55/100. But if it asks for the probability given that the student is in Year 11, you are confined to the Year 11 row, so it is 32/50.
That word given is the whole question. It tells you to shrink the denominator from the grand total to a single row or column. Missing it is the most common error in this topic.
Relative frequency is the same calculation applied to observed data rather than theory: the proportion of times something actually happened. The SAT uses it to ask whether a sample's results can be extended to a wider population, which they can only when the sample was selected at random.
Worked examples
Using the table above, what is the probability a randomly chosen student travels by car? The denominator is the grand total, because no restriction was given
P(car)=10045=0.45
What is the probability that a student travels by car, given that they are in Year 12?
A27/100
B27/50
The word given restricts you to the Year 12 row, whose total is 50. So the answer is 27/50, making (B) correct. Choice (A) is the trap of keeping the grand total as the denominator.
In a random sample of 200 shoppers, 46 bought fruit. The shop serves 5000 customers a week. Estimate how many buy fruit. The relative frequency is 20046=0.23, and because the sample was random it can be applied to the whole population
0.23×5000=1150
Had the sample not been random, no such estimate would be justified — and that is usually the correct answer choice when it appears.
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