I started with the easy part: each guess is correct with probability 0.5, so a streak of 10 has base probability (0.5)^10.
Model each player's sequence of guesses as independent Bernoulli trials with success probability p, and define a 'ban' as the occurrence of a run of 10 consecutive successes within 100 trials. Derive the probability of such a run using recurrence relations or Markov chains, then multiply by N for the expected number of banned players. Discuss approximations like Poisson or normal for large N, and clarify assumptions about p and independence.
Pro tip: Always state your assumptions explicitly (e.g., independence, constant p) and mention that the expected value is linear regardless of dependence between players. This shows you understand the underlying probability theory and can communicate it clearly.
State that each guess is an independent Bernoulli trial with success probability p, and that a ban occurs if there is a run of 10 consecutive successes within 100 trials. Define the indicator variable for each player being banned.
Use a recurrence relation or Markov chain to compute P(ban) = probability of at least one run of length 10 in 100 trials. For example, let f(n) be the probability of no run of 10 in n trials, and derive a linear recurrence.
By linearity of expectation, the expected number of banned players is N * P(ban). This holds regardless of dependence between players.
For small p, approximate P(ban) using Poisson approximation or the expected number of runs. For large N, the distribution of banned players is approximately normal or Poisson, depending on P(ban).
Present the final expression, note any simplifying assumptions (e.g., p=0.5), and mention that the result can be validated via simulation or exact computation for small n.
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