ACT Math: Calculate Probability of Compound Events Using Tree Diagrams

Formula 1: P(A and B) when events are independent. Multiply: P(A)×P(B). Example: Probability of rolling a 3, then rolling a 3 again is (1/6)×(1/6)=1/36. Formula 2: P(A or B) when events are mutually exclusive. Add: P(A)+P(B). Example: Probability of rolling a 3 OR a 4 on one roll is (1/6)+(1/6)=2/6=1/3. Formula 3 (Complex): P(A or B) when events can both happen (not mutually exclusive). Subtract overlap: P(A)+P(B)-P(A and B). Example: P(card is red or Ace)=P(red)+P(Ace)-P(red Ace)=(26/52)+(4/52)-(2/52)=28/52. Always ask yourself: Are these events independent, dependent, or mutually exclusive? The answer determines which formula to use.

A tree diagram visualizes compound events. First branch shows outcome of event 1 (e.g., coin flip: heads or tails). Second branch shows outcome of event 2. The path probability is the product of all branches on that path.

Error 1: Multiplying when you should add. P(rolling a 3 or a 4) ≠ (1/6)×(1/6). Error 2: Forgetting to account for overlap. P(A or B)≠P(A)+P(B) if events can both occur. Error 3: Confusing conditional probability P(B|A) with compound probability P(A and B). Error 4: Assuming events are independent when they're not. Drawing without replacement changes the second probability. Always read the problem carefully: "with replacement" means independent; "without replacement" means dependent.

Quick fix: If unsure, draw a tree diagram. It forces you to think through each step and catches errors early.

Problem 1: Flip a coin twice. P(two heads)=(1/2)×(1/2)=1/4. Problem 2: Draw two cards from a deck without replacement. P(first is Ace and second is King)=(4/52)×(4/51)=16/2652≈1/166. Problem 3: P(rolling a die and getting 2 OR rolling a 5)=(1/6)+(1/6)=2/6=1/3. Complete all three, then swap a detail (e.g., "with replacement" vs. "without") and redo to see how the answer changes.

Bonus: P(at least one head in two flips). It's easier to find 1-P(no heads)=1-(1/2)×(1/2)=1-1/4=3/4, than to add P(HT)+P(TH)+P(HH). Try both approaches and verify they match.

ACT Math includes 1-2 compound probability questions, usually medium difficulty. They test your understanding of independence, overlap, and basic counting. Students who master the two formulas can solve these questions in two minutes and move forward without anxiety.

Spend 15 minutes this week solving 6-8 compound probability problems. By test day, you'll recognize independent vs. dependent scenarios instantly and choose the right formula every time.