(Odds) Flush vs. Overflush

This is my true story not very long ago.

My hole cards were Kd 4d in SB. By turn, there were three diamonds on board. Happy with my K-high flush, I raised 1/2 pot and some other guy reraised all in all of a sudden. I called to see his Ad Qd... DOH! (Good it is not real money.)

Case K-High: King-high flush runs into Ace-high flush

There are at least 3 suited cards on the board. Maybe 4, maybe 5.
We have to list them all.

- 3 suited cards on board (no Ace):
You hold 2. There are 7 non-Ace cards left.
Therefore, there are 7 kinds of A-high flush out there.
Total possibility of hole cards = 1225

In a field of 9 opponent, none of them has A-high flush: (1-7/1225)^9 = 0.9497
Therefore, chance of bumping into A-high flush = 1-0.9497 = 0.0503 (1 in 20)

To summarize: (3/4/5 flush cards on board)
9 opponents: 1 in 20/23/28
7 opponents: 1 in 25/30/35 <-- This is what happened to me!
5 opponents: 1 in 35/41/49
3 opponents: 1 in 59/68/82
1 opponent: 1 in 175/204/245

Case Q-High: Queen-high flush runs into King-high flush or above

- 3 suited cards on board (no Ace or King, if there is any then go back to above example)
You hold 2. There are 7 A-high flush, 6 K-high flush:

1-((1-13/1225)^9 = 0.0916 (1 in 11)

To summarize: (3/4/5 flush cards on board)
9 opponents: 1 in 11/13/16
7 opponents: 1 in 14/16/20
5 opponents: 1 in 19/23/28
3 opponents: 1 in 32/37/46
1 opponent: 1 in 94/111/136

Case J-High: Jack-high flush runs into Queen-high flush or above

- 3 suited cards on board (no Ace, King or Queen. If there is any go back to above example)
You hold 2. There are 7 A-high flush, 6 K-high flush, 5 Q-high flush:

1-((1-18/1225)^9 = 0.124 (1 in 8.0)

To summarize: (3/4/5 flush cards on board)
9 opponents: 1 in 8.0/9.5/12
7 opponents: 1 in 10/12/15
5 opponents: 1 in 14/17/21
3 opponents: 1 in 23/28/34
1 opponent: 1 in 68/82/102

Case T-high: Ten-high flush runs into Jack-high flush or above

Note: Still, if there is one of A,K,Q or J on board, go back to the previous case. If there are two, go back two levels, etc.

To summarize: (assume 3 flush cards on board)
9 opponents: 1 in 6.6
7 opponents: 1 in 8.4
5 opponents: 1 in 12
3 opponents: 1 in 19
1 opponent: 1 in 56

Case 9-high: 9-high flush runs into Ten-high flush or above

To summarize: (assume 3 flush cards on board)
9 opponents: 1 in 5.9
7 opponents: 1 in 7.4
5 opponents: 1 in 10
3 opponents: 1 in 17
1 opponent: 1 in 49

As you can see, your chance of being a flush underdog becomes quite significant when your own flush isn't very good, while against a large table. If you have an 8-high flush, chances are that about 1 in 8 times you are entitled to lose all your chips.

A few days earlier, in one of our 4-player home games, a guy had 56 flush. There were 2 over-cards on board (so he has an equivalent 8-high flush). His chance of doom was 1-((1-27/1225)^3, about 1 in 15. Indeed, he ran into an K-high flush and lost his third buy-in!

hey thanks for running the numbers on this i always wonder about odds like this but im numerically chalenged (I'm an english major)

This happened to me just now. Should have just checked the river.