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The requirements for a binomial experiment are well known (see, for example, http://www.stat.yale.edu/Courses/1997-98/101/binom.htm):

  1. Fixed number of trials (n),
  2. Each trial has two outcomes (success and failure),
  3. Probability of success is constant (p),
  4. 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).

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  • $\begingroup$ It depends on the exact meaning of the requirement. If the requirement 3 means "Probability of success is constant (p) regardless of the other trial outcomes", then it implies 4. If we interpret the statement "Probability of success is constant (p)" as $$\mathrm{Pr}(x_i=success) = p,$$ Then this is about the marginal probability of success. It is still possible that the conditional probability is different from $p$. Then, the requirement 4 is needed. $\endgroup$
    – Kota Mori
    Oct 28, 2022 at 5:05
  • $\begingroup$ An explanation of marginal probability can be found here: towardsdatascience.com/…. I don't see how to rewrite the Rugby/Football/Male/Female example such that it makes sense here. Could you please provide a simple example? $\endgroup$
    – Christopher Donham
    Oct 28, 2022 at 12:20

1 Answer 1

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

  1. Fixed number of trials -- Yes, observations of "n" people.
  2. Two outcomes -- Yes, "person saw heads" or "person did not see heads".
  3. Probability constant -- Yes, P(Success) = 0.5
  4. Independent -- No.
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  • $\begingroup$ This example is somewhat trivial in that the dependence is complete, with a correlation of 1 $\endgroup$
    – kjetil b halvorsen ♦
    Nov 1, 2022 at 0:43

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