odds on hitting a straight by the river, starting with connectors?

[university]

The odds of hitting a flush when you hold 2 suited cards, and staying all the way to the river, are 16.3 to 1 against.

Found that on this site http://casinogambling.about.com/library/weekly/aa102802a.htm

I'm looking to find information on what the odds of hitting a straight are, when you hold 2 connected cards (suited or unsuited) and you stay all the way to the river.

For example, you hold 6,7. What are the odds, that you'll hit a straight after all cards are on the table?

Thanks for any information....

[Freudian]

There is a around ~6.4% chance to get flush and a ~8.5% chance of a straight, when you play suited connectors (that can get the full range of straights, ie 54s-JTs). This includes the rare situations where you are using all 5 cards on the board for your flush/straight.

[BugsBunny]

I really hope someone checks these numbers.

The number given (16.3 to 1 against) assumes that only a 3 flush will be dealt in the 5 board cards. This is calculated as follows:

Number of combinations of 3 flushes of your suit:
C(11,3) = 165
that leaves 47 remaining cards in the deck, of which 39 are not of the flush suit. We need 2 more cards to make up the remainder of the board. So:
C(39,2) = 741

The total number of 5 card boards with a 3 flush = 165 * 741 = 122265

The total number of possible 5 card boards = C(50,5) = 2118760

122265 / 2118760 = 0.0577059223 = 5.77% or 16.33 to 1 against.

If you include the possibility of 4 and 5 flushes being dealt then instead of C(39,2) you would use C(47,2), which equals 1081. That gives:
165 * 1081 / 2118760 = 0.0841836735 = 8.42% or 10.88 against

If you want to look at just 3 or 4 flushes then subtract out the number of 5 flush boards. C(11,5) = 462
(165 * 1081 - 462) / 2118760 = 0.0839656214 = 8.4 % or 10.91 against

Now with a straight assume a maximal connector like TJ. That means that you have the following possibilities for a straight:
789
89Q
9QK
QKA

There are 4 sets of 3 cards. Each set has 4*4*4 combinations so you get 64 combinations for each set for a total of 64 * 4 = 256 combinations.
Allowing for the possibilty of 1 of ranks in your hand also being dealt out we're back to C(47,2). That gives us:
256 * 1081 / 2118760 = 0.1306122449 = 13.06% or 6.66 to 1 against. This compares to the 10.88 to 1 against number for a flush.

For the chance of your straight coming up with no duplication of ranks in your hand then instead of 47 cards remaining in the deck we use 41.
C(41,2) = 820.
256 * 820 / 2118760 = 0.0990768185 = 9.91% or 9.09 to 1 against

If we want to look at the chance of a maximally suited connector completing either the straight or the flush then we have to make some adjustments since some of the straight combinations may also be flushes.

straight flushes = 4 total

2 cards of the straight are flush cards and at least one other card is a flush card.
9 combinations per set have exactly 2 of your flush cards. 4 sets so 36 total combinations. 47 cards remaining with 9 of those being flush cards. So 36 cards aren't flush cards.
C(36,2) = 630 combinations with no flush cards.
C(47,2) = 1081
1081 - 630 = 451 combinations which include at least 1 flush card.
36 * 451 = 16236 = total number of combinations where you have 2 flush cards as part of your straight and at least 1 other flush card appears.

1 card of the straight is a flush card and the other 2 board cards complete the flush.
27 combinations per set for a total of 108 combinations where the straight cards have exactly 1 flush card.
10 flush cards left in the remaing 47 cards.
C(10,2) = 45 combinations with 2 flush cards.
108 * 45 = 4860

Total straights that are also flushes = 4 + 16236 + 4860 = 21100

(256 * 1081 - 21100) / 2118760 = 0.1206535898 chance of getting a straight with no flush.

total chance of getting a straight or a flush when holding a maximally suited connector and staying to the river:
0.1206535898 + 0.0841836735 = 0.2048372633 = 20.48% or 3.88 to 1 against

[university]

Bruce Z mentions in this post Odds to flop a draw (http://archiveserver.twoplustwo.com/showflat.php?Cat=&Board=&Number=278584&page=0&view =expanded&sb=5&o=14&fpart=1) that the chance of just flopping either a flush or straight draw is around %19.3 percent. Which is similar to the number you came up with, Bugs, and your number reflects going to the river....

Something seems off.

[BugsBunny]

I'll look at it again sometime later, but it might not be as off as you think. Remember that the hard part in getting either the straight or the draw is flopping the 1st 3 cards to match your hand. Once you have that it's more likely that you'll connect on the turn or river.

So the possibilities of making the straight or flush will be heavily weighted towards the possibilities of getting a draw.

With all that said it's very possible that I'm off somewhat and may have missed something along the line. Which is why I said that I hope somebody checks the numbers.

[Randy Burgess]

I'm going to stick my neck out and say those odds for making a flush in your suit w/2 suited hole cards if you stay to the river are wrong. "Hold'em Odds Book" by Mike Petriv gives the percent chance of this as 6 percent, which would give odds against of 15.6 percent, but I believe Petriv is rounding.

A more exact calculation goes like this:

Calculate ways to miss a flush by adding ways to make a 5-card board with one of your suit, with two of your suit, and with none of your suit:

(39 4)*(44 1) + (39 3)*(11 2) + (39 5) = 1,983,163

Total ways to make a five-card board not including your hole cards are (50 5) or 2,118,760.

So your ways to make a flush = Total ways to make the board - total misses = 135,597.

This gives odds against of 14.63, and percentage chance of making your flush as roughly 6.3998.