The requirements for a binomial experiment are well known (see, for example, http://www.stat.yale.edu/Courses/1997-98/101/binom.htm):
- Fixed number of trials (n),
- Each trial has two outcomes (success and failure),
- Probability of success is constant (p),
- Trials are independent.
Counter examples that are unique to the first 3 requirements are easily generated.
Question: Using language appropriate for an introductory statistics class, could someone please provide an explanation and example of something that meets the first 3 requirements (fixed number of trials, two outcomes, probability of success is constant), but fails only the 4th requirement (trials are independent)?
Discussion: I am used to the test for independence meaning P(A|B) = P(A) and P(B|A) = P(B) (see https://link.springer.com/referenceworkentry/10.1007/978-0-387-31439-6_744).
Applying that to this situation, I believe that means that the probability of success in trial 2 given failure in trial 1 is P(Success) = p -- true because of the constancy requirement. I also need probability of failure in trial 2 given success in trial 1 is P(Failure) = (1 - p) -- again, true because of the constancy requirement. So it seems like the probability being constant implies independence. What am I missing? (again, language and example appropriate for an introductory statistics class please).