Question 1155350
Let's look at spades, as an example:
P(1st card is a spade) = 13/52 = 1/4
P(2nd is a spade) = 12/51
P(3rd is a spade) = 11/50
P(4th is a spade) = 10/49
P(5th is a spade) = 9/48
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P(all 5 cards are spades) = (13/52)(12/51)(11/50)(10/49)(9/48) = 154440/311875200
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Since we are looking for any flush, just multiply the above by 4 (essentially, the first card is a freebie, that's why I simplified the first fraction to 1/4):
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(12/51)(11/50)(10/49)(9/48) = 11880/5997600 = {{{ green(0.00198) }}}  or about 1 in 505 hands.

This answers the question as asked; however, this INCLUDES straight-flushes (where the rank of the cards are in consecutive order such as 5-6-7-8-9 of hearts).  If you want an answer that does not include traight-flushes in the count of ordinary flushes then note there are 40 straight-flushes in 52C5 poker hands (works out to a probability of 0.00001539).  Subtracting this quantity from the first highlighted answer gives P(true flush) = {{{ highlight( 0.00196 ) }}}, or about 1 in 509 hands.