Probability measures how likely an event is. SAT questions in this chapter focus on identifying the correct sample space, deciding whether one event changes another, and translating AND or OR language into the appropriate operation.
Outcomes, trials, events, and sample spaces
| Term | Meaning | Original example |
|---|---|---|
| Trial | One performance of a random process | Spin a fair spinner once |
| Outcome | One possible result of a trial | The spinner lands on B |
| Sample space | All possible outcomes under consideration | \(\{A,B,C,D\}\) |
| Event | One or more outcomes of interest | Landing on A or C |
Use this ratio only when the listed outcomes are equally likely.
Count favorable outcomes
A fair spinner has eight equal sectors labeled \(1\) through \(8\). What is the probability of spinning a factor of \(8\)?
- List favorable outcomes
The factors shown are \(1,2,4,8\), so there are \(4\) favorable outcomes.
- Divide by the sample space
There are \(8\) equally likely sectors, so \(P=4/8=1/2\).
Independent and dependent events
Does the first event change the second?
Independent
The first result does not change the probability of the second. Standard draws with replacement are independent.
Dependent
The first result changes the remaining counts or probabilities. Standard draws without replacement are dependent.
Replacement determines dependence
A two-panel comparison shows how returning or keeping the first item changes the second draw.
What happens after the first draw?
AND means follow one complete path
Use the unchanged second probability when the events are independent.
The second factor must reflect what remains after event \(A\).
Two draws with replacement
A container has 3 green tokens and 2 gold tokens. Each token is returned after selection, so every second-stage branch keeps the original probabilities.
Without replacement
A container holds \(5\) square tiles and \(3\) round tiles. Two tiles are drawn without replacement. What is the probability of square, then round?
- First draw
The probability of square is \(5/8\).
- Update totals
After removing a square, \(7\) tiles remain and all \(3\) round tiles remain.
- Multiply the path
\(P(\text{square then round})=\frac58\cdot\frac37=\frac{15}{56}\).
OR for mutually exclusive events
- Mutually exclusive events
- Events that cannot occur on the same trial. One outcome cannot belong to both events.
This chapter-level rule applies when \(A\) and \(B\) cannot occur together. For a finite equally likely sample space, direct counting of distinct favorable outcomes is also reliable.
Add disjoint outcomes
A fair number cube is rolled. What is the probability of rolling a \(1\) or a \(6\)?
- Check overlap
A single roll cannot be both \(1\) and \(6\), so the events are mutually exclusive.
- Add
\(P(1\text{ or }6)=1/6+1/6=2/6=1/3\).
Same-category multi-draw problems
Calculate each valid branch
- Name the branches
For two categories, 'same category' means first-first or second-second.
- Multiply each AND path
Update the second probability if sampling is without replacement.
- Add the disjoint paths
The two complete paths cannot happen simultaneously, so add their probabilities.
Two tiles of the same shape
A box has \(4\) triangles and \(2\) circles. Two shapes are drawn without replacement. What is the probability that they match?
- Triangle branch
\(P(TT)=\frac46\cdot\frac35=\frac25\).
- Circle branch
\(P(CC)=\frac26\cdot\frac15=\frac1{15}\).
- Add branches
\(\frac25+\frac1{15}=\frac7{15}\).
Recognition guide
| Prompt signal | Action | Check |
|---|---|---|
| AND, then, followed by | Multiply along one path | Does the first event change the second? |
| OR between mutually exclusive events | Add the disjoint event probabilities | Can both occur on one trial? |
| With replacement | Restore original counts | Second denominator stays the same |
| Without replacement | Update remaining counts | Second denominator decreases by one |
Common mistakes and traps
- Adding probabilities for an AND path instead of multiplying.
- Multiplying mutually exclusive alternatives instead of adding them.
- Treating no-replacement draws as independent.
- Changing the second denominator after an item was replaced.
- Reducing the total but forgetting to reduce the favorable category after drawing from it.
- Counting overlapping favorable outcomes twice in an OR question.
- Reporting a favorable-outcome count instead of dividing by the sample-space size.
Replacement check
A jar has \(3\) mint and \(5\) peach tokens. Two are drawn without replacement. What is \(P(\text{mint then mint})\)?
- \(\frac38\cdot\frac38\)
- \(\frac38\cdot\frac27\)
- \(\frac38+\frac27\)
- \(\frac28\cdot\frac37\)
Show answer and explanation
Answer: \(\frac38\cdot\frac27=\frac3{28}\)
After one mint token is removed, \(2\) mint tokens remain among \(7\) total tokens.
What to remember
- For equally likely outcomes, probability is favorable outcomes divided by total outcomes.
- Replacement restores the original composition; no replacement changes it.
- Multiply probabilities along an AND path.
- Add probabilities for mutually exclusive OR alternatives.
- For same-category questions, calculate each valid branch and then add the branches.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.