🎯 Learning Objectives
- The single probability formula and what counts as a 'favorable outcome'.
- How to read a two-way (frequency) table to extract any probability quickly.
- Conditional probability: 'given that' problems.
- Independent vs. dependent events: how to multiply correctly.
- 10 SAT-style practice questions.
1. The Universal Probability Formula
P(event) = favorable outcomes / total outcomes
Every basic probability question reduces to this single fraction. The hardest part is identifying what counts as 'favorable' and what counts as 'total': the rest is arithmetic.
⚡ Strategy: Just Count, Don't Compute
A probability can be any real number from 0 to 1. A fraction such as 47/93 can be valid. Check the event count and the sample space rather than judging an answer by its denominator.
2. Two-Way (Frequency) Tables
A two-way table cross-classifies the data by two variables. SAT probability questions love them because they let the test ask many different probability questions from a single table.
Sample Table
Likes Coffee | Doesn't Like Coffee | Total | |
Adults | 60 | 20 | 80 |
Teens | 10 | 30 | 40 |
Total | 70 | 50 | 120 |
Three Probability Question Types from One Table
Question Wording | Total (Denominator) | Favorable (Numerator) |
Probability of being a teen who likes coffee | Grand total (120) | Cell value (10) |
Probability that a teen likes coffee | Teens row total (40) | Cell value (10) |
Among coffee lovers, probability of being a teen | Coffee column total (70) | Cell value (10) |
💡 Pro Tip: Watch the Word 'Given'
If the question says 'given that' or 'among', it's a conditional probability: the denominator changes from the grand total to a row or column total.
3. Independent vs. Dependent Events
❓ Does the first event affect the second? | ||
❌ NO (independent) ▼ | ✅ YES (dependent) ▼ | |
Multiply: P(A and B) = P(A) × P(B) | Use updated probabilities (e.g., one fewer item) | |
📝 Worked Example: Drawing Without Replacement
A bag has 4 red and 6 blue marbles. You draw two without replacement. What is the probability both are red?
First draw: P(red) = 4/10.
Second draw (now 9 marbles, 3 red): P(red) = 3/9.
Combined: 4/10 × 3/9 = 12/90 = 2/15.
4. Or, And, and Not
Logic | Formula | Example |
P(not A) | 1 − P(A) | P(not red) = 1 − P(red) |
P(A or B), mutually exclusive | P(A) + P(B) | P(red or blue) = P(red) + P(blue) |
P(A and B), independent | P(A) × P(B) | Two coins both heads = 1/2 × 1/2 |
5. Common Pitfalls
⚠️ Common Mistakes
- Confusing 'P(A given B)' with 'P(A and B)'. Conditional uses a smaller denominator.
- Multiplying probabilities for events that are NOT independent.
- Forgetting that 'at least one' often equals 1 − P(none).
- Counting an outcome twice when adding ('or' for non-mutually-exclusive events).
6. Strategies
⚡ Use complements for 'at least one'
P(at least one) = 1 − P(none). Way faster than adding cases.
⚡ For two-way tables, identify the denominator first
Grand total, row total, or column total: pick before counting.
⚡ Convert percentages to fractions
30% = 3/10 makes mental math instant.
⚡ Sketch a tree diagram for two-step problems
Multiplying along branches and adding across paths is foolproof.
⚡ Sanity-check: probability is between 0 and 1
If your answer is 1.4 or −0.3, something is wrong.
7. Summary
📌 Key Takeaways
- Basic probability: favorable / total.
- Conditional probability: denominator narrows to the 'given' group.
- Independent: P(A and B) = P(A) × P(B).
- Dependent: Update probabilities after each step.
- Complement: P(not A) = 1 − P(A).
- 'At least one': use 1 − P(none).
8. Practice: 10 Questions
Q1. A bag has 5 red and 7 blue marbles. P(red)?
A) 5/12 B) 7/12 C) 5/7 D) 1/2
Hint: Favorable / total = 5/12.
Q2. A standard die is rolled. P(even number)?
A) 1/6 B) 1/3 C) 1/2 D) 2/3
Hint: 3 evens out of 6 = 1/2.
Q3. A table shows 30 boys and 20 girls. P(student is a girl)?
A) 1/3 B) 2/5 C) 1/2 D) 3/5
Hint: 20/50 = 2/5.
Q4. Two coins are flipped. P(two heads)?
A) 1/8 B) 1/4 C) 1/2 D) 3/4
Hint: 1/2 × 1/2 = 1/4.
Q5. A bag has 3 red and 5 blue. Two are drawn without replacement. P(both red)?
A) 1/8 B) 3/28 C) 1/4 D) 3/14
Hint: 3/8 × 2/7 = 6/56 = 3/28.
Q6. P(at least one head in two coin flips)?
A) 1/4 B) 1/2 C) 3/4 D) 1
Hint: 1 − P(no heads) = 1 − 1/4 = 3/4.
Q7. From a class of 100 with 60 boys and 40 girls (25 boys play sports, 15 girls play sports), P(plays sports)?
A) 1/4 B) 3/10 C) 2/5 D) 1/2
Hint: (25 + 15) / 100 = 40/100 = 2/5.
Q8. Same class. Among sports players, P(girl)?
A) 15/40 B) 15/100 C) 40/100 D) 25/40
Hint: 15 girls / 40 sports players = 3/8.
Q9. P(rolling a 5 on a six-sided die OR flipping heads)?
A) 7/12 B) 1/12 C) 1/3 D) 1/2
Hint: 1/6 + 1/2 − (1/6 × 1/2) = 1/6 + 1/2 − 1/12 = 7/12.
Q10. A drawer has 4 black and 6 white socks. You grab one without looking, replace it, and grab again. P(both white)?
A) 9/25 B) 3/5 C) 1/4 D) 6/10
Hint: Independent: 6/10 × 6/10 = 9/25.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | A) 5/12 | 5 reds out of 12. |
Q2. | C) 1/2 | 3 evens out of 6. |
Q3. | B) 2/5 | 20/50. |
Q4. | B) 1/4 | 1/2 × 1/2. |
Q5. | B) 3/28 | 3/8 × 2/7. |
Q6. | C) 3/4 | 1 − P(no heads) = 3/4. |
Q7. | C) 2/5 | 40/100. |
Q8. | A) 15/40 = 3/8 | Conditional on sports players. |
Q9. | A) 7/12 | 1/6 + 1/2 − 1/12. |
Q10. | A) 9/25 | (6/10)² with replacement. |