Common Mistakes
Traps in Probability
5 mistake patterns students fall for. Each one shows the wrong approach vs the correct approach.
Confusing independence with mutually exclusive
FORMULAMutually exclusive events (A∩B = ∅) with non-zero probabilities are DEPENDENT, not independent.
✗ WRONG: A and B are mutually exclusive, so P(A∩B) = P(A)·P(B)
✓ RIGHT: If A∩B = ∅, then P(A∩B) = 0 ≠ P(A)·P(B) (when both > 0). They are dependent.
Wrong sample space in conditional probability
FORMULAWhen given a condition, the sample space changes. Forgetting this leads to wrong answers.
✗ WRONG: Computing P(A|B) as P(A) without restricting to B
✓ RIGHT: P(A|B) = P(A∩B)/P(B). First find the intersection, then divide by P(B).
Binomial: confusing n and r
FORMULAIn P(X=r) = nCr·p^r·q^(n-r), n is the number of trials and r is the number of successes.
✗ WRONG: Swapping p and q, or using wrong n
✓ RIGHT: p = probability of success in ONE trial. n = total trials. r = desired successes. q = 1-p.
With/without replacement confusion
CASE MISSWith replacement: probabilities stay same (independent). Without: probabilities change each draw.
✗ WRONG: Treating without-replacement draws as independent
✓ RIGHT: Without replacement: use conditional probabilities or combinations. With replacement: multiply same probabilities.
Variance formula sign error
FORMULAVar(X) = E(X²) - [E(X)]², not E(X²) - E(X). The mean must be squared.
✗ WRONG: Var(X) = E(X²) - E(X)
✓ RIGHT: Var(X) = E(X²) - [E(X)]². Compute E(X) first, square it, then subtract from E(X²).