I play poker occasionally and have been thinking about the probability to make an all-red (diamond, heart) or all-black (club, spade) hand – a monochromatic hand. What I have come up with seems like the correct answer and I would like to get confirmation or correction.
According to Wikipedia the probability of hitting a flush is $508:1$. So I would think that it is as simple as $254:1$, since in a regular flush you can only use one suit but here you can use two.
I am pretty sure there is something I am missing, since if you have $4$ hearts you make a flush with another heart, but a monochromatic with a diamond.
There are $\binom{26}5$ all-red hands and the same number of all-black hands. There are $\binom{52}5$ hands overall. The probability of drawing a monochrome hand is $$2\frac{\binom{26}5}{\binom{52}5}\approx0.0462.$$