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Let S be the number of spades left in the deck.
Let C be the number of cards to come.
Let D be the number of cards Hero can not see (left in the deck).
The chance of drawing EXACTLY 1 more spade in this round:
(S,1)*(D-S,C-1)/(D,C)
where the notation (a,b) means "a choose b". E.g. (4,2) = 6
Then the odds of making your flush is 1 minus the odds you you drawing 0 spades.
1 - (S,0)*(D-S,C)/(D,C)
But we know that ANYTHING choose ZERO is always 1. So it reduces to:
1 - (D-S,C)/(D,C)
That is the equation to tell you the % chance you will complete your flush.
Hand 1:
Assumptions:
1) You are first to act, so there are 40 cards that you can not see (because they are in the deck).
* If you are 2nd to act, then there would be 39.
Givens:
1) There are 7 spades left in the deck
S = 7
2) There are 7 cards left to be dealt.
C = 7
3) There are 40 cards left in the deck.
D = 40
So your chance of making the flush, if you do NOT place the 9 there now is:
1 - (D-S,C)/(D,C)
1 - (40-7,7)/(40,7)
1 - (33,7)/(40,7)
1 - 4,272,048/18,643,560
1 - 0.2291
0.771
77.1%
Hand 2:
Basically the same as Hand 1, except there is 1 less "spade" in the deck. Here, we can just replace "spade" with diamond.
S = 6, C = 7, D = 40
1 - (D-S,C)/(D,C)
1 - (40-6,7)/(40,7)
1 - (34,7)/(40,7)
1 - 0.2885
0.711
71.1%
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