HOLD 3 CARDS
The only time that you hold 3 cards is if you have a 3-of-a-kind or a 3-card-straight-flush, 3-card-royal-flush, or 3 face cards.
Hold 3 Card Straight Flush
I did not list the straight flush odds that the hold cards contained an Ace, 2 or 3 because of space. Here is why.
Are all 1 in 1081 odds
Are all 1 in 540 odds
Is 1 in 360 odds
In the first group (1081 odds),
If the set has an A the SF can only be A, 2, 3, 4, 5
If the set has a 2 and a 6 it can only be 2,3,4,5,6
If the set has a 3 and a 7 it can only be 3,4,5,6,7
In the second group (540 odds)
Each straight flush can be completed by adding one card on each end or one inside card and one end card.
The SFs possible are A,2,3,4,5 2,3,4,5,6 3,4,5,6,7
In the third set (360 odds)
Two cards on either end of the set can be 1, 2 or 6, 7
Or one card on each end of the set a 2 and 6.
All complete the Straight Flushs.
1,2,3,4,5 2,3,4,5,6 or 3,4,5,6,7
Hold 3 Face Cards
If the three face cards do not contain an ace they must be J, Q, K. If the three cards are suited the odds of catching a straight are less because some of the cards caught would result in a SF or RF.
Straight / no ace
If the three face cards are suited a 9-10 suited or 10-ace would result in a SF or RF and would not be counted as a straight. This is why the odds of catching a straight when holding unsuited face cards is 1 in 34 and the odds are 1 in 36 when the three face cards are suited.