$$ \begin{align*} Y =& \begin{cases} 1 & \text{ if "heads"} \\ 0 & \text{ if "tails"} \end{cases} \\ P(Y=1) =& w \\ \hat Y \equiv& 1 \end{align*} $$

$$ \begin{align*} \text{SSE} =& \sum\limits_{i=0}^{n-1}\left(y_i - \hat y_i\right)^2 & \text{SST} =& \sum\limits_{i=0}^{n-1}\left(y_i - \bar y\right)^2 \\ =& \sum\limits_{i=0}^{n-1}\left(y_i - 1\right)^2 & =& \sum\limits_{i=0}^{n-1}\left(y_i - 0.5\right)^2 \\ =& \sum\limits_{i=0}^{n-1}\left(y_i^2 -2y_i + 1^2\right) & =& \sum\limits_{i=0}^{n-1}\left(y_i^2 - 2(0.5)y_i + 0.5^2\right) \\ =& \sum\limits_{i=0}^{n-1}\left(-y_i + 1\right) & =& \sum\limits_{i=0}^{n-1}\left(0.25\right) \\ =& -nw + n & =& 0.5n \\ =& n(1-w) & =& 0.5n \\ \end{align*} $$

So, our $R^2$ is:

$$ \begin{align*} R^2 =& 1 - \frac{\text{SSE}}{\text{SST}} \\ =& 1 - \frac{n(1-w)}{0.5n} \\ =& 2w - 1 = 1.1 - 1 = 0.1 \end{align*} $$

Richard C. Grinold, Ronald N. Kahn