Confusing independence with mutually exclusive
Very CommonCONCEPT
Mutually exclusive events (A∩B=∅) with non-zero probabilities are DEPENDENT, not independent.
Why: Both concepts involve two events, but independence means P(A∩B)=P(A)⋅P(B), while mutual exclusion means P(A∩B)=0.
WRONG: A and B are mutually exclusive, so P(A∩B)=P(A)⋅P(B) RIGHT: If A∩B=∅, then $P(A \cap B) = 0
eq P(A) \cdot P(B)(whenboth> 0$). They are dependent. Wrong sample space in conditional probability
Very CommonCONCEPT
When given a condition, the sample space changes. Forgetting this leads to wrong answers.
Why: Students compute P(A) over the original sample space instead of restricting to B.
WRONG: Computing P(A∣B) as P(A) without restricting to B RIGHT: P(A∣B)=P(B)P(A∩B). First find the intersection, then divide by P(B). See pattern: Bayes' Theorem →Variance formula sign error
Very CommonFORMULA
Var(X)=E(X2)−[E(X)]2, not E(X2)−E(X). The mean must be squared.
Why: Forgetting to square E(X) is the most common algebraic slip in probability.
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). Binomial: confusing n and r
CommonFORMULA
In P(X=r)=nCr⋅pr⋅qn−r, n is the number of trials and r is the number of successes.
Why: The formula has four variables (n,r,p,q) and it is easy to swap p and q or use wrong n.
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. See pattern: Binomial Distribution →With/without replacement confusion
CommonCASE MISS
With replacement: probabilities stay same (independent). Without: probabilities change each draw.
Why: Students default to multiplication of same fractions even when the pool size changes after each draw.
WRONG: Treating without-replacement draws as independent
RIGHT: Without replacement: use conditional probabilities or combinations. With replacement: multiply same probabilities.
See pattern: Conditional Probability →