Probability and Probability Distributions
What the exam tests
This domain tests sample spaces, event operations, addition and multiplication rules, conditional probability, and binomial or Poisson probabilities. The exam frequently uses without-replacement selection and “at least” wording.
Core rules
- (P(A\cup B)=P(A)+P(B)-P(A\cap B)). For mutually exclusive events, the intersection term is zero.
- (P(A|B)=\frac{P(A\cap B)}{P(B)}), (P(B)>0). For independent events, (P(A\cap B)=P(A)P(B)).
- Binomial: (P(X=r)={}^nC_r p^r q^{n-r}), with (q=1-p), mean (np), variance (npq).
- Poisson: (P(X=r)=\frac{e^{-\lambda}\lambda^r}{r!}), mean and variance both (\lambda).
- For “at least one”, use (1-P(\text{none})).
Worked example
A fair die is rolled 4 times; let (X) be the number of sixes. Then (p=\frac16), (q=\frac56), (n=4). (P(X=2)={}^4C_2\left(\frac16\right)^2\left(\frac56\right)^2=6\times\frac1{36}\times\frac{25}{36}=\frac{150}{1296}=\frac{25}{216}). The binomial model applies because trials are independent with constant (p).
Traps
Mutually exclusive and independent are different ideas. Without replacement changes the denominator and often the probability on the next draw. The binomial requires fixed (n), two outcomes, constant (p) and independence. A Poisson mean is not the same as a binomial mean unless the context says so. Check whether “at least” includes the endpoint.