medium Probability
A release-checked medium problem for training Discrete Random Variables, Games, Calculus.
You are making a biased coin, where you can decide the probability $p$ of obtaining a heads. What value of $p$ maximizes the probability that you obtain exactly two consecutive heads in $4$ flips? $$$$ For example, $HHHT$ and $HHHH$ are not allowed, but $HHTH$ is allowed. Your answer will be in the form $\dfrac{a-\sqrt{b}}{c}$ for integers $a,b,$ and $c$ with $c$ minimal. Find $a + b + c$.
This Probability problem is tagged Discrete Random Variables, Games, Calculus. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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