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The profitability of a play in poker is determined on the basis of the risk vs. reward concept. This concept takes a simple mathematical form in the definitions of various odds and the relations between them. The most common use of odds is the comparison of drawing odds and pot odds.

A drawing hand is a hand that has little value in the current situation but can improve significantly through certain future cards (outs) which are said to complete a draw. Drawing odds are defined as the ratio of the probability that a hand will not complete the draw to the probability that a hand will complete a draw, calculated as \(\frac1-pp\), where \(p\) equals the probability of completing the draw. They are commonly stated in the form of a fraction \(\left(\frac1p - 1\right):1\).

Flush Draw

The hand \( A\diamondsuit J\diamondsuit \) on the board of \(2\clubsuit 6\diamondsuit 8\diamondsuit T\clubsuit \) has no immediate value. However, if a card that is a diamond comes, the hand will turn into a flush. The probability that the next card is a diamond, ignoring the possible holdings of opponents, is \(p=\frac946\approx 19.6 \% \) (there are 9 diamonds out of 46 unseen cards). Drawing odds of completing the flush in this situation are approximately \(4.11:1\).

Pot odds are defined as the ratio \(\fracPC\) of the current size of the pot \((P)\) to the cost of the call \(C\).

You are playing against one opponent who bets \($10\) in a pot of \($20\). The current size of the pot is \($30\). If you call, you are paying \($10\) to win \($30\) so your pot odds are \(3:1\).

The expected value of the call with a drawing hand where no future actions are possible is determined by the relation of drawing odds to pot odds:

Expected value of the call with a drawing hand with no future actions:

Suppose that a player is holding a hand that has the probability \(p\in (0,1) \) to complete a draw, thus becoming the winning hand and the player has to call a bet of size \(b\) to win a pot of size \(P\). The expected value of the call is calculated as \(E = p\times P - b\times (1-p)\) \((\)if the hand completes the draw the win is \(P\), otherwise the loss equals the cost of the call \(b).\) The call has positive expected value \(E>0\) if and only if

\[ p\times P - b\times (1-p)>0 \iff \frac1-pp
that is, if the pot odds are numerically larger than the drawing odds.

Calling with a Draw

You hold \(9\diamondsuit 6\heartsuit \) and the flop is \(A\diamondsuit 7\spadesuit 8\heartsuit\). Your opponent has a stack of half of the pot size and moves all in. He coincidentally flips his cards over, revealing \(J\diamondsuit J\clubsuit\). Do you have proper pot odds to call his all in?

Your probability of winning the hand if you call his all in is around 33%, or expressed as odds, roughly 2:1. If you call his half-pot sized all in bet, your pot odds are 3:1. Since you are getting better pot odds than winning odds, the call has a positive expected value. \(_\square\)

The confrontation of a made hand (a hand with no significant chances of improving) and a drawing hand illustrates the most common application of the concept of odds. A player with a drawing hand compares the odds to determine if the call is profitable while a player with a made hand can adjust the size of the bet to deny favorable odds to a drawing hand, making the call unprofitable.

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