Step 1: Read and classify the problem

• "Repeated independent trials with fixed p" → Binomial
• "Given that X occurred" → Conditional probability
• "Which source/cause" → Bayes' theorem
• "At least one" → Complement method
• "How many ways / equally likely" → Classical counting

Step 2: Set up the calculation

• Binomial: identify n, p, and the desired range of r
• Conditional: identify P(AB) and P(B), compute ratio
• Bayes': identify partition, priors, likelihoods
• Classical: count favorable and total using C or P

Step 3: Compute

• Use complement when direct computation has many cases
• For binomial "at least k," compute 1 - sum of terms below k
• For Bayes', always compute denominator (total probability) first

Step 4: Verify

• Check P is between 0 and 1
• Check all probabilities in a distribution sum to 1
• For binomial: exponents sum to n, coefficient is C(n,r)
• For Bayes': all posterior probabilities should sum to 1

Common time-savers:

• P(at least one) = 1 - P(none) [much faster]
• For p = 1/2: P(X=r) = C$\frac{n,r}{2}$^n [symmetric]
• q = $\frac{Var}{Mean}$ to find binomial parameters quickly