I must change your question to this:
>>What are the chances of being dealt a hand that will lose to a 2-3-4-5-7 flush, the lowest possible flush?<<
That's because the way you have it stated,
>>What are the chances of being dealt a hand that will lose to a flush?<<,
the answer would depend on the flush that you are talking about, for a higher
flush beats a lower flush. For instance, a queen-high flush beats a jack-high
flush or any other flush with a lower high card.
First we'll find the number of possible hands that are
(1) flushes (straight or not),
(2) full houses,
or
(3) 4-of-a-kind's
Then we'll subtract from the total number of poker hands with 5 cards, which is
C(52,5). Then we'll divide by C(52,5).
Flushes and straight flushes
The total number of flushes and straight flushes is C(4,1)ŚC(13,5) because for
each of the C(4,1) ways to choose the 1 suit, there are C(13,5) ways to choose
the 5 denominations. That's C(4,1)ŚC(13,5) = 5148
The number of full houses is C(13,1)C(4,3)C(12,1)C(4,2) because for each of the
choices of denomination for the three that are alike, we choose 3 suits for
them. Then for each of the choices of the 12 remaining denomination for the two
that are alike, we choose 2 suits from them.
C(13,1)C(4,3)C(12,1)C(4,2) = 3744
The number of 4-of a kind's is C(13,1)ŚC(48,1) because for each of the ways you
can choose the denomination of the 4, there are 48 ways to choose the 5th card.
That's C(13,1)ŚC(48,1) = 624
Therefore the total number of flushes (straight or not), full houses, and
4-of-a-kind's are 5148 + 3744 + 624 = 9516
The total number of 5-card hands are 2598960, so the number of hands that will
lose to the lowest possible flush are 2598960-9516 = 2589444
So the chances of being dealt a hand that will lose to the lowest possible
flush is 
which reduces to 
= .9963385354.
Round that however you are told.
Edwin
