Probability and conditional probability
Problem-Solving and Data Analysis is about 15% of SAT Math (5–7 of 44 questions). Probability questions usually come with a two-way table or a short story, and the most common type asks for a conditional probability: “given that…”.
The basics
If all outcomes are equally likely, the probability of an event is the number of favorable outcomes divided by the number of possible outcomes. It is always between 0 (impossible) and 1 (certain). The probability that an event does not happen is 1 minus the probability that it does. If you repeat something many times, the expected number of times an event happens is its probability times the number of repetitions: a probability of 0.2 in 50 tries is expected about 10 times.
Two-way tables
A two-way table sorts the same group in two ways at once. The last row and last column are totals. The inside cells count people with both properties. Below, 100 students are sorted by grade and club. A student is picked at random. The probability of chess is 45/100. The probability of “grade 10 and chess” is the single cell, 18/100.
| Chess | Music | Total | |
|---|---|---|---|
| Grade 10 | 18 | 22 | 40 |
| Grade 11 | 27 | 33 | 60 |
| Total | 45 | 55 | 100 |
“Given that”: conditional probability
The words “given that” or “if the student is in…” tell you to look at only one row or column. That group becomes the whole: it is the new denominator. P(A | B) = (number with A and B) ÷ (number with B). In the table, P(chess | grade 10) = 18/40 = 0.45, because only the 40 students of grade 10 count. The reverse, P(grade 10 | chess) = 18/45 = 0.4, is a different question with a different denominator.
| Chess | Music | Total | |
|---|---|---|---|
| Grade 10 | 18 | 22 | 40 |
| Grade 11 | 27 | 33 | 60 |
| Total | 45 | 55 | 100 |
Independent events
Two events are independent if knowing one gives no information about the other: P(A | B) = P(A). Then P(A and B) = P(A) × P(B). In the table, P(chess) = 0.45 and P(chess | grade 10) = 0.45 = 27/60 = P(chess | grade 11), so the club does not depend on the grade. Two flips of a coin are independent: the chance of two heads is 1/2 × 1/2 = 1/4. Picking two items from a bag without putting the first back is not independent, because the first pick changes what is left.
“Or”, “not” and “at least one”
For “A or B” add the two probabilities and subtract the overlap, so that outcomes in both are not counted twice. For “at least one”, it is usually easier to find the probability of none and subtract from 1. For example, the probability of at least one head in 3 flips of a fair coin is 1 − (1/2)³ = 7/8.
| Tea | Coffee | Total | |
|---|---|---|---|
| Under 30 | 20 | 22 | 42 |
| 30 or older | 30 | 8 | 38 |
| Total | 50 | 30 | 80 |
- 8/30
- 22/80
- 22/42
- 22/30
- “Who prefers coffee” is the condition, so only the Coffee column counts: 30 people.
- Of these, 22 are under 30.
- P = 22/30 = 11/15, about 0.73.
- First marble red: 5/8.
- Now 4 red marbles are left among 7: second red: 4/7.
- Multiply: 5/8 × 4/7 = 20/56 = 5/14.
- “At least one 6” is the opposite of “no 6 at all”.
- P(no 6 on one roll) = 5/6, so P(no 6 on both rolls) = (5/6)² = 25/36.
- P(at least one 6) = 1 − 25/36 = 11/36.
- The 8 children who do both are inside both counts, so subtract them once.
- Swim or cycle: 24 + 18 − 8 = 34 children.
- P = 34/40 = 0.85.
- Using the wrong denominator“Given that B” makes the group B the whole. Divide by the total of B, not by the grand total.
- Reversing the conditionP(A | B) and P(B | A) are different. In “given that she plays chess”, the condition is chess.
- Adding probabilities for “and”For “and” of independent events, multiply. For “or”, add and subtract the overlap.
- Double counting in “or”Outcomes in both groups are inside both counts. Subtract them once.
- Treating draws without replacement as independentAfter the first pick the group is smaller. Reduce both the favorable count and the total.
- Answers outside 0 to 1A probability can never be negative or greater than 1. Use this to check your answer. Type it as a fraction (3/8) or a decimal (0.375).
Desmos works as a calculator for probabilities. Type 5/8 * 4/7 and Desmos shows the decimal value. It cannot read a table for you, so first decide which row or column counts and which numbers go on top and bottom. Use it to check products such as (5/6)^2, or to compare a decimal with a fraction choice.
- Type 5/8 * 4/7 to see 0.357…
- Type 5/14 on the next line: it shows the same decimal, so 5/8 · 4/7 = 5/14
- Type 1 − (5/6)^2 and 11/36: both show 0.305…
Use Desmos for long products or to turn a fraction into a decimal. Choosing the right group (the denominator) is a thinking step that no calculator can do.
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