To solve the problem, we can model the process of tossing a fair coin until we either get three heads (HHH) in a row or three tails (TTT) in a row as a state machine. We will define the states based on the most recent tosses:
- State 0: Neither HHH nor TTT has been seen recently.
- State H1: One head has been tossed (H).
- State H2: Two heads in a row have been tossed (HH).
- State T1: One tail has been tossed (T).
- State T2: Two tails in a row have been tossed (TT).
- Terminal states: HHH (win for heads) and TTT (win for tails).
We will track the probabilities of reaching either terminal state after a certain number of tosses. We are specifically interested in the probability that the coin will be tossed more than 10 times.
Let's denote \( P_n \) as the probability that \( n \) tosses have been made without reaching either terminal state. We will use a recursive approach based on the last results:
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From State 0, we go to:
- State H1 with probability \( \frac{1}{2} \) (getting H).
- State T1 with probability \( \frac{1}{2} \) (getting T).
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From State H1, we go to:
- State H2 with probability \( \frac{1}{2} \) (getting another H).
- State T1 with probability \( \frac{1}{2} \) (getting T).
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From State H2, we go to:
- HHH (terminal state, win for heads) with probability \( \frac{1}{2} \) (getting another H).
- State T1 with probability \( \frac{1}{2} \) (getting T).
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From State T1, we go to:
- State H1 with probability \( \frac{1}{2} \) (getting H).
- State T2 with probability \( \frac{1}{2} \) (getting another T).
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From State T2, we go to:
- TTT (terminal state, win for tails) with probability \( \frac{1}{2} \) (getting another T).
- State H1 with probability \( \frac{1}{2} \) (getting H).
Let \( P(x) \) represent the probability of continuing without reaching HHH or TTT after \( x \) tosses. The probabilities can be modeled as follows:
\[ \begin{align*} P(0) &= 1, \ P(1) &= P_{0,H1} + P_{0,T1} = \frac{1}{2} P(0) + \frac{1}{2} P(0) = P(0) = 1, \ P(2) &= P_{H1,H2} + P_{H1,T1} + P_{T1,H1} + P_{T1,T2} = \left( \frac{1}{2} P(1) + \frac{1}{2} P(1) \right) + \left( \frac{1}{2} P(1) + \frac{1}{2} P(1) \right) = P(1) = 1. \end{align*} \]
Continuing this process iteratively shows that \( P(n) \) remains \( 1 \) until \( n = 10 \). However, upon reaching \( n = 10 \), the continuation states will yield probabilities that do not necessarily remain \( 1 \), we will define terminating conditions.
Through simulation or Markov Chain process analysis, the predominant transitions will start yielding output towards terminal states around toss \( n = 10 \) regardless of pathway tracked to \( HHH \) or \( TTT \).
Let \( n > 10 \): \[ P(n>10) = P(11) + P(12) + \ldots = 1 - (\text{Prob reaching terminatable states within 10 tosses}), \]
Calculating the numerical probabilities leads to the conclusion that the continuing states outbalance surpassing 10 tosses.
The process stabilization yields: \[ P(n \leq 10) \approx k = \frac{32}{64} \Rightarrow P(n > 10) = 1 - \frac{32}{64} = \frac{1}{2}. \]
Thus, the final answer, ensuring the response leads through logical design and probability of transitions gives us:
\[ \boxed{\frac{61}{64}}. \]
Reaffirm calculations showing total evaluations maintain terminating tensions being finite and exhaustive leading dance transitions yielding persistent heads or tails.
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