How Far Is a Coin-Flip Count from Uniform?

A fair coin is flipped twice and $X$ counts the heads, so $X \sim \text{Binomial}(2,\ 0.5)$ over the outcomes $\{0, 1, 2\}$. Call this true distribution $P$.

A model $Q$ wrongly assumes the three outcomes are equally likely, i.e. $Q(0) = Q(1) = Q(2) = 1/3$.

Compute the Kullback–Leibler divergence in bits:

$D_{KL}(P \parallel Q) = \sum_{k} P(k)\log_2\frac{P(k)}{Q(k)}$

Useful values: $\log_2 3 \approx 1.5850$, $\quad \log_2 0.75 \approx -0.4150$, $\quad \log_2 1.5 \approx 0.5850$.