Odds versus probability

Distinguish wins-to-losses odds from a share of all outcomes.

7 minutes · beginner · Original Poker4Quant lesson

What you will learn

Convert odds against an event into probability and identify an ambiguous odds statement.

Prerequisites: Chips, fractions and percentages

The denominator tells you what is being compared

Probability compares favorable outcomes to all outcomes. Odds in favor compare favorable outcomes to unfavorable outcomes. If there are 2 favorable and 3 unfavorable equally likely outcomes, the probability is 2/5 while odds in favor are 2:3.

Odds against reverse that comparison: 3:2 against. The phrase “4 to 1” needs context—does it describe odds against, odds in favor, or a payoff ratio? Do not convert it until you know what its two numbers mean.

Add the two groups

If odds against are a:b, there are a unfavorable units for every b favorable units. The probability is b/(a+b). For 4:1 against, that is 1/(4+1) = 20%, not 25%.

A payoff ratio is a price, not a measured success rate. Being offered 4 chips of net profit per 1 chip risked does not establish how often the event occurs. Later lessons compare a price-implied threshold with a separately modeled probability.

Worked example

A simplified deck contains 6 winning cards and 18 losing cards. Express the chance of a win as probability and odds against.

All possible cards total 6 + 18 = 24.
Probability is 6/24 = 1/4 = 25%.
Odds against compare losses to wins: 18:6 = 3:1.

25% probability and 3:1 odds against describe the same event. The odds denominator excludes losing outcomes; the probability denominator includes them.

Common mistakes

Turning 3:1 against into 1/3 instead of 1/4.
Confusing payout odds with an observed win rate.
Dropping the words “in favor” or “against.”

Keep these ideas

Probability is favorable divided by total.
Odds compare favorable and unfavorable groups.
State which direction the odds refer to before converting.