There are 10 possible sequences that can be straights, because the ace can be
considered as the lowest card of {A,2,3,4,5} or the highest card of {10,J,Q,K,A}.

Those 10 sequences are:

{A,2,3,4,5}, {2,3,4,5,6}, {3,4,5,6,7}, {4,5,6,7,8}, {5,6,7,8,9}, 
{6,7,8,9,10}, {7,8,9,10,J}, {8,9,10,J,Q}, {9,10,J,Q,K}, {10,J,Q,K,A}

First let's just find the number of all-red straights.

For each card in each of those 10 sequences, there are 2 choices for a red suit,
hearts or diamonds.  That's 25=32 choices for their suits.
So there are 10∙32=320 all-red hands of cards in sequence.

However that 320 counts two red straight flushes (where the suits are all the same).
We eliminate those by subtracting 2.  320-2 = 318

There are also 318 all-black straights. So that's a total of 2∙318 = 636 same color straights.    

So 636 is the numerator of the desired probability.

The denominator is the number of possible 5-card poker hands.
That's 52 cards CHOOSE 5 = 52C5 = 2,598,960.

So the probability that you will be dealt a same color straight (but not all
the same suit) is



Edwin