The Handy Math Answer Book

In five-card pokerthe probability of drawing a royal flush (a poker hand with the

ace, king, queen, jack, and 10 all in the same suit) is 1.54  10 ^6 , with odds of

649,739:1. For a straight flush (a poker hand with consecutive cards in the same suit,

but not a royal flush), the probability is 1.39  10 ^5 , with odds of 72,192.3:1. Three of

a kind (three cards with the same value) has a probability of 0.0211, with odds of

46.3:1, while one pair (two cards of the same value) has a probability of 0.423, with

odds of 1.366:1.

In bridge13 top honors has a probability of 6.3  10 ^12 and odds of 158,753,389,

899:1. A 12-card suit, ace high, has a probability of 2.72  10 ^9 , and odds of

367,484,697.8:1. Getting four aces has a probability of 2.64  10 ^3 , with odds of

377.6:1.

What is the casino’s or house edge?
A casino is a gambling facility that normally includes all or a combination of the fol-
lowing: slot machines, video games, card games, and other games such as keno, craps,
and bingo. In order to make the casino the major money-maker—not the gambler—
there are certain “rules” of the casino, including the casino’s edge, otherwise known
as the house edge.
The house edge is the ratio of the expected player loss to the initial amount bet; it
is an exact measurement, usually expressed as a percent, of the casino’s advantage in a
432 game. A casino earns money by paying winners at “house odds,” an amount that is

What are the possible numbers of distinct five-card hands

in five-card poker and thirteen-card hands in bridge?

T

here is, of course, a mathematical way of determining the possible number of

distinct five-card hands in poker—or the various combinations of five cards

from a 52-card deck. The formula, in which Nis the number of combinations, is

as follows (note: the 52 written over 5 is called a binomial coefficient; for more

about coefficients, see “Algebra”):

Not to be outdone, such a number can be determined for bridge games, too.

In the following, Nagain represents the combinations:

N 13 ,,,combinations

52

==^9 C 635 013 559 600

N==5,,2 598 760combinations

952 C