# Probabilities and Odds

Throughout the discussion that follows and in the mathematical charts and tables, references will be made to the probabilities of events occurring and the odds against events occurring. Judging by the mail I received following the first edition of this book, the two terms—probabilities and odds—are frequently confused. Probabilities are related to odds, but these quantities are defined differently.

Probabilities are expressed either with a number between zero and one, or as a percentage. A probability of zero means the event will never happen, while a probability of one means that the event is a certainty. When the probability is between zero and one, that means the outcome is uncertain. Consider the common occurrence of having four cards of the same suit (a four-flush) with one card to come. There are nine remaining cards that complete the flush out of 46 that have not been seen (you see the two cards in your hand and four on the board). You will make a flush 9 out of every 46 times this situation happens, or about one-fifth of the time, and not make the flush four-fifths of the time. The probability for a flush can be expressed as one-fifth, as its decimal equivalent 0.2, or as 20%. The probability for not making a flush is four-fifths, or 0.8, or 80%. Note that these combined probabilities add to one, or 100%, because it is certain that you will either make the flush or not.

Odds are expressed ratios. The odds are the average number of failures for each success. For the example of the flush, in which success occurs 20% of the time, there will be four failures on average for each success. That means the odds against making the flush are 4 to 1. If the probability of an event is 50%, then there is one success on average for every failure. In this case, the odds against success are 1 to 1.