You know a straight flush beats a full house and the odds of landing that inside straight in Texas Hold 'Em with just the river card left are worse than 10-1. But how good are you with the numbers, really? Robert Hannum, the DU professor whose poker research is the subject of this week's cover story, "All In," furnished us with the following brain teasers. Test your chops, then check the answers (and a few handy hints for doing the calculations yourself) below.
1. In Texas Hold'Em, the game starts (after necessary blind bets and/or antes) with two cards dealt face down to each player. How many different two-card combinations are possible?
2. After a round of betting, three community cards called the flop are turned face-up on the table. How many different flops are possible (excluding your own two hole cards)? Note: The relevant consideration is the number of unseen cards; the number of players is not relevant to this question.
3. In Texas Hold'Em, what is the best possible hand (including hole cards) and second best possible hand (independent of the best hand, so cards may be "re-used") if the board is:
a) A♠ 10♥ 8♣ 8♥ 7♥ b) 5♠ 6♣ 9♥ 6♦ Q♣
4. In Texas Hold'Em, what are the odds against making a flush with two cards to come when you hold two suited cards, and two cards of your suit appear on the flop?
5. A single card is randomly selected from a standard deck of 52 playing cards.
(a) What is the probability the selected card is a red Ace? (b) What is the probability the selected card is red or an Ace? (c) If it's a face card, what's the probability that it's a King? (d) If it's a King, what's the probability it's a face card? (e) If it's red, what's the probability that it's an Ace?
EXTRA CREDIT. Some dice questions:
6. A pair of honest dice is thrown once.
(a) What is the probability the total is seven? (b) What is the probability the total is seven or eleven? (c) What is the probability the total is two, three, or twelve? (d) What is the probability the total is a four, five, six, eight, nine, or ten?
7. A casino is offering a dice game where the only wager is on rolling snake-eyes (both dice showing one) with a payoff of 30 to 1. What is the house edge for this game?
3. (a) Best: Jack-high straight flush (J♥ 9♥)
Second Best:10-high straight flush (9♥ 6♥)
(b) Best: 4 sixes
Second Best: Queens full of sixes
4. The probability of making the flush can be calculated as follows:
P(make Flush) = 1 - [(38/47)x(37/46)] = 1 - .650 = .350
Odds: .650/.350 = 1.86 to 1
5. (a) 2/52 = 1/26 = 0.038
(b) 28/52 = 7/13 = 0.538
(c) 4/12 = 1/3 = 0.333
(d) 4/4 = 1.00 (no flies on you!)
(e) 2/26 = 1/13 = 0.077
6. (a) 6/36 = 1/6 = 0.167
(b) 8/36 = 2/9 = 0.222
(c) 4/36 = 1/9 = 0.111
(d) 24/36 = 2/3 = 0.667
7. The house advantage is 13.9 percent. Why? Because the expected value to the player is a negative .139:
EV = (+30)(1/36) + (-1)(35/36) = -0.139
For more on the math behind the odds, see Hannum's online Guide to Casino Math.