This is the board : Board: [9d 8h 7c]
My hand : [10d 7s]
Opponents (unknown): [Xx Xx]
Cards remaining in the deck is 45. As there is still a turn and river to come, the total combination of hands is (n, k) = (45,2) = 990 combinations.
Outs to improve hand: {J, 6, 10, 7} = (4+4+3+2) = 13 outs.
Is the following calculation of combining probabilities correct? It seems a rather large percentage.
Method 1:
17.154 + 17.154 + 13.017 + 8.78 = 56.105 to improve hand on turn and river.
Method 2:
170 + 170 + 129 + 87 = 556 / 990 = 56.16 to improve hand on turn and river.
Calculations:
[J] every [J] (=4) will complete a straight. Probability of catching a [J] on the turn and river is the objective. 52-7 = 45 cards left (4 [J]’s, 41 others)
Method 1
P([J] on turn OR river)
P([J] on turn) + P([J] on river) - P(both)
4/45 + 4/45 - ((choose(4,2)/choose(45,2))
8.88 + 8.88 - 0.606 = 17.154 %
Method 2
Determining number of hands
Any [J] will do, that is 4 cards.
Two [J]’s combinations: (n, k) = (4, 2) = 6 combinations
One [J] combinations: 4*41 = 164
Total = 170
170/990*100 = 17.17
[6]: every [6] (=4) will complete a straight as well. Probability of catching a [6] on the turn and river is the objective. 52-7 = 45 cards left (4 [6]’s, 41 others)
Method 1
P([6] on turn OR river)
P([6] on turn) + P([6] on river) - P(both)
4/45 + 4/45 - ((choose(4,2)/choose(45,2))
8.88 + 8.88 - 0.606 = 17.154%
Method 2
Determining number of hands
Any [6] will do, that is 4 cards.
Two [6]’s combinations: (n, k) = (4, 2) = 6 combinations
One [6] combinations: 4*41 = 164
Total = 170
170/990*100 = 17.17
[10]: as one [10] is taken, there are 3 [10]’s left in the deck. A [10] on the turn or river will give two pair and thus hand improvement: [10d] [10x] = [10d] [10x] [7s] [7c]
Method 1
P([10] on turn OR river)
P([10] on turn) + P([10] on river) - P(both)
3/45 + 3/45 - ((choose(3,2)/choose(45,2))
6.66 + 6.66 - 0.303 = 13.017%
Method 2
Determining number of hands
Only 3 [10] will do.
Two [10] combinations: (n, k) = (3, 2) = 3 combinations
One [10] combinations: 3*42 = 126
Total = 129
129/990*100 = 13.03%
[7]: as two [7]’s are taken, there are 2 [7]’s left in the deck. A [7] on the turn or river will give 3 of a kind and thus hand improvement: [7x] = [7s] [7c] [7x]
Method 1
P([7] on turn OR river)
P([7] on turn) + P([7] on river) - P(both)
2/45 + 2/45 - ((choose(2,2)/choose(45,2))
4.44 + 4.44 - 0.303 = 8.78%
Method 2
Determining number of hands
Only 2 [7]’s will do.
Two [7]’s combinations: (n, k) = (2, 2) = 1 combination
One [7] combinations: 2*43 = 86
Total = 87
87/990*100 = 8.78%
