odds

Odds express the ratio of favorable to unfavorable outcomes, while probability expresses the proportion of favorable outcomes out of all possible outcomes.

For an event with probability $p$:

• Odds = $\frac{p}{1-p}$
• Probability = $\frac{\text{odds}}{1+\text{odds}}$

Rolling a 6 on a standard die

• Probability: $\frac{1}{6}\approx 0.167$
• Odds: $\frac{1/6}{5/6}=\frac{1}{5}$ or “1 to 5”

This means for every 1 time you roll a 6, you expect 5 times you won’t.

Advantages over probabilities

Multiplicative symmetry: If event A has odds 2:1, then not-A has odds 1:2. With probabilities: 2/3, 1/3

Natural for updates: In Bayes Theorem, updating beliefs multiplies odds by the likelihood ratio:

$$\text{posterior odds}=\text{prior odds}\times \text{likelihood ratio}$$

This is simpler than the probability form which requires normalization.

Unbounded scale: Probabilities are confined to [0,1], but odds range from 0 to ∞. This makes them natural for log-odds - symmetric around 0, unbounded in both directions.

Betting interpretation: “3 to 1 odds” directly tells you the payoff structure - bet $1 to win $3.

Odds appear naturally in logistic regression where we model log-odds as a linear function of predictors. They’re also central to odds ratios for comparing risks between groups.