Details on Two Statements by Sklansky

[brianmarc]

1. In TOP he says odds of flopping a set are "about 8/1". What is the math for this?

2. In HEFAP he provides a table of probabilites for completing a hand by the river. But he doesn't provide the math. Example: For 9 outs the probability of completing the hand is 35%. Again, what is the math for this?

[Mano]

For making a set (or quads) on the flop the easiest way is to calculate the odds of not making it and subtracting from 1. Of the 50 cards in the deck other than your pair, 48 will not make you a set so the probability of not making a set or quads are (48/50)*(47/49)*(46/48) = .882 so the odds of making it are 1 - .882 = .118. To convert to odds .882/.118 = 7.5:1 against.

For 9 outs we can solve the same way. With 2 cards to come there are 47 possible cards, 9 of which makes your hand, so the probability of not making your hand is (38/47)*(37/46) = .65 so the probability of making it is 1-.65 = .35 which is 1.86:1 against.

[bigpooch]

1) Odds of flopping a set (or quads):

C(m,n) = the number of combinations of m objects chosen n at
a time. For almost all purposes, m>=n and m and n are
nonnegative integers.

E.g., C(52,5) = number of draw poker hands and C(52,2) =
number of holdem hands. It is calculated by the formula

C(m,n) = (m!)/[((m-n)!)(n!)] where k! denotes the product
1 x 2 x...x k.

If you have a pocket pair, there are now only 50 cards to
choose from and three are chosen at random to make up the
flop; therefore, there are C(50,3) = 19600 possible flops.

Quads: 48 possible combinations (choose the two case cards
that match the rank and any other)

Exactly trips: 2 (matching rank) x C(48,2) (any two of the
other) = 2 x 1128 = 2256

Thus, the probability of making exactly trips is 2256/19600
or about 0.1151020 or 7 97/141 to 1 against or about
7.687943 to 1 against.

Including quads, the proability increases to 2304/19600 or
about 0.1175510 or 7 73/144 to 1 against or about 7.506944
to 1 against which is about 7.5 to 1 against.

2) Probability of completing a hand with k outs where k>=1.

Actually, the table could have been extended from k=1 to
23 to be complete. Since there are 47 unknown cards, and if
k were bigger than 23, just take j=47-k and look the number
for j on the table and subtract the percentage from 100%.

The simple reason is you are just looking at outs for the
competition. And of course, for k=0, the answer is trivial!
And in LHE, you only need a very small portion of this
table!

One easy way to compute the table is consider the chances of
missing instead. There are 47 unseen cards and therefore
C(47,2)=1081 possible combinations of turn and river cards.

If there are k "key cards" to make the hand, there are then
47-k cards that miss. There are simply C(47-k,2) such
combinations of two "misses". Thus, the probability of a
miss is C(47-k,2)/C(47,2) and obviously if this is
subtracted from 1, the probability of hitting is found.

Strangely, in my edition of HPFAP, the entry for 1 out is
4.4% but a calculation shows C(46,2)/C(47,2) = 0.9574468 so
that the last entry should read 4.3% instead. All the
entries for the common situations are correct.

[Lost Wages]

For making a set (or quads)...

You calculation also includes full houses.

Lost Wages

[Mano]

You calculation also includes full houses.

So it does. I guess I should have said flopping trips or better (was ignoring impact of cards which don't match your hole pair in rank).

[bigpooch]

Of course, as previously mentioned, the numbers for "exactly
trips" also include full houses!

[Homer]

Both of these questions are discussed in detail in the archives.

[brianmarc]

Simple. Elegant. Thank you.

[brianmarc]

Thank you for the tutorial. I now am armed to do the work I need.

[brianmarc]

For 9 outs: There are 47 possible cards, 9 of which makes your hand, so the probability of not making your hand is (38/47)*(37/46) = .65 so the probability of making it is 1-.65 = .35"

This can be generalized to 1-((47-A17)/47)*((46-A17)/46) for N outs.