Requirements for Binomial Distributions 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).

Select n people.  Toss a coin.  Success is "person saw heads".  Requirements:

*

*Fixed number of trials -- Yes, observations of "n" people.

*Two outcomes -- Yes, "person saw heads" or "person did not see heads".

*Probability constant -- Yes, P(Success) = 0.5

*Independent -- No.

