Probability - BSEB Inter Study Guide
Probability
This domain tests conditional probability, independence, Bayes theorem, random variables and the binomial distribution. Draw a tree diagram for two-stage experiments: multiply along a branch and add across branches.
Core rules
- P(A|B)=P(A∩B)/P(B), with P(B)>0.
- Multiplication rule: P(A∩B)=P(A)P(B|A). For independent events, P(A∩B)=P(A)P(B).
- Bayes theorem: P(A_i|B)=P(A_i)P(B|A_i) divided by the sum of P(A_j)P(B|A_j).
- For a binomial variable X~B(n,p): P(X=r)=nCr p^r q^(n−r), mean=np, variance=npq.
- Mutually exclusive events cannot occur together; independent events do not affect each other's probabilities.
Worked example
A bag has 2 white and 3 black balls. Two balls are drawn without replacement. Find P(both white). P(first white)=2/5. Given first white, 1 white remains among 4 balls, so P(second white|first white)=1/4. Thus P(both white)=2/5×1/4=1/10.
Traps
- Using with-replacement probabilities when the question says without replacement.
- Confusing P(A|B) with P(B|A).
- Treating mutually exclusive events as independent.
- Using the binomial formula when trials are not independent or n is not fixed.
- Forgetting that total probability in a Bayes problem must sum to 1.