Calculating the expected height of a randomly built binary search tree

I need to calculate the expected height of a randomly built binary search tree, BST, with 4 different keys: $x < y < z < w$

According to Catalan numbers, there are 14 possible trees, 8 with height 3 and 6 with height 2. So let $X$ be a random discrete variable that holds the tree height, $P{X=2}=frac{6}{14}=frac{3}{7}$ and $P{X=3}=frac{8}{14}=frac{4}{7}$ hence $E[X]=2cdot frac{3}{7} + 3cdotfrac{4}{7} = frac{18}{7}approx 2.571$

But then I thought, not all the trees in Catalan numbers has the same probability of occurring as there are 24 ways to order the keys, so instead I get $P{X=2}=frac{16}{24}=frac{2}{3}$ and $P{X=3}=frac{8}{24}=frac{1}{3}$ and $E[X]=2cdot frac{2}{3} + 3cdotfrac{1}{3} = frac{4}{3} + 1 = frac{7}{3} approx 2.333$

I’m not sure which way is the correct way to calculate the probability of the possible values for $X$