SAT Probability & conditional probability
Single events and two-way tables — including conditionals.
How to score it
- Probability = favorable ÷ total — the trick is choosing the right total.
- “Given that …” or “of the [subgroup]” restricts the denominator to that row/column.
- Read the table's margins carefully; the right total is often a subtotal.
Common traps
- Uses the grand total when the question restricts to a subgroup.
- Swaps numerator and denominator.
- Misreads which row or column the condition names.
The 4 question types, with real examples
Two-way table probability
“If a [unit] is chosen at random from [subset], what is the probability it [attribute]?”
The table summarizes responses to a survey question by employment status for 300 respondents.
. If one respondent who answered Yes is selected at random, what is the probability that the respondent is employed?Employed: Yes 85, No 65, Total 150 Unemployed: Yes 40, No 60, Total 100 Retired: Yes 20, No 30, Total 50 Total: Yes 145, No 155, Total 300
- A
- B✓
- C
- D
Given that the respondent answered Yes, we restrict to the 145 respondents who said Yes. Of these, 85 are employed. The conditional probability is 85/145 = 17/29.
“What fraction of X is Y”
“What fraction of the [restricted group] [has attribute]?”
In a class of 120 students, 72 students study Spanish, 48 students study French, and 24 students study both Spanish and French. If one student is selected at random, what is the probability that the student studies French but not Spanish?
- A✓
- B
- C
- D
There are 48 students who study French, and 24 of them also study Spanish. Therefore, 48 - 24 = 24 students study French but not Spanish. The probability is 24/120 = 1/5.
Single-event uniform probability
“[n-sided die / n objects in a bag]. What is the probability of [event]?”
A bag contains 18 red balls, 24 blue balls, and 30 green balls. If one ball is selected at random, what is the probability that the ball is not green?
- A✓
- B
- C
- D
The total number of balls is 72. The number of balls that are not green is 18 + 24 = 42. The probability is 42/72 = 7/12.
Compound probability (multi-stage)
“[n trials, each with probability p]. [question]”
A jar contains 15 marbles: 5 are green, 6 are blue, and 4 are red. If two marbles are selected at random without replacement, what is the probability that both marbles are blue?
- A✓
- B
- C
- D
The probability that the first marble is blue is 6/15. After one blue marble is selected, there are 5 blue marbles left out of 14 total marbles. The probability that the second marble is also blue is 5/14. The probability that both are blue is (6/15) × (5/14) = 30/210 = 1/7.
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