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Found problems: 2

1974 Putnam, A4
An unbiased coin is tossed $n$ times. What is the expected value of $|H-T|$, where $H$ is the number of heads and $T$ is the number of tails?
expected value coin tosses probability

2021 Miklós Schweitzer, 10
Consider a coin with a head toss probability $p$ where $0 <p <1$ is fixed. Toss the coin several times, the tosses should be independent of each other. Denote by $A_i$ the event that of the $i$-th, $(i + 1)$-th, $\ldots$ , the $(i+m-1)$-th throws, exactly $T$ is the tail. For $T = 1$, calculate the conditional probability $\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)$, and for $T = 2$, prove that $\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)$ has approximation in the form $a+ \tfrac{b}{m} + \mathcal{O}(p^m)$ as $m \to \infty$.
conditional probability probability coin tosses