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Part of 2021 Miklós Schweitzer

Problems(1)

Asymptote of a conditional probability in a coin toss

Source: 2021 Miklos Schweitzer, P10

11/2/2021
Consider a coin with a head toss probability pp where 0<p<10 <p <1 is fixed. Toss the coin several times, the tosses should be independent of each other. Denote by AiA_i the event that of the ii-th, (i+1)(i + 1)-th, …\ldots , the (i+m−1)(i+m-1)-th throws, exactly TT is the tail. For T=1T = 1, calculate the conditional probability P(A2ˉA3ˉ⋯Amˉ∣A1)\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1), and for T=2T = 2, prove that P(A2ˉA3ˉ⋯Amˉ∣A1)\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1) has approximation in the form a+bm+O(pm)a+ \tfrac{b}{m} + \mathcal{O}(p^m) as m→∞m \to \infty.
probabilityconditional probabilitycoin tosses