10 months ago Mathe...

What are the odds? Part - 2 of Math Probability

The biggest difficulty in the way of calculation of probability is that for complex systems, data is usually insufficient and we do not know all parameters of the functional system. Hence the concept of subjective logic- a branch of probability that deals with uncertain parameters and allows assumptions in calculations.

 

Probability has implications more than just mathematical – philosophers find that a fundamental aspect of the human condition is that nobody can ever determine with absolute certainty whether a proposition about the world is true or false. Also, individual people can never judge large systems objectively- they allow their assumptions of the world into their judgements. There is even a branch of probability called “fuzzy logic”, wherein there are no hard and fast answers to questions, the “yes” and “no” collide. Hence we have much larger questions- is there a certain universal truth? Can individuals ever be unbaised enough to think ??/completely objectively? It is for the next generation to find out. :)

Frequently Asked Questions (FAQs)

  • Probability measures the chance of an event happening, expressed as a fraction from 0 to 1 (or 0%–100%).
  • Odds, however, compare the chance of occurrence to non-occurrence: odds \( for = P/(1–P) \), and odds \( against = (1–P)/P \)

So while probability gives a direct likelihood, odds provide a comparative ratio.

  • From probability to odds (in favor):  \( Odds = P / (1 – P) \) 
  • From odds to probability:  \( P = odds / (1 + odd) \)

For example, if P = 0.25, odds = 0.25 / 0.75 = 1:3. If odds are 3, probability = 3 / (1+3) = 0.75.

In experiments with equally likely outcomes (Bernoulli trials), odds are the ratio of successes to failures: S : F

Then P = S / (S + F) and odds_for = S / F. Useful for simple events like dice or coin tosses.

  • Odds help in betting, statistics, and logistic regression, offering insight into likelihood relative to non-occurrence.
  • They remain valid even with extreme values (close to 0 or 1), while probabilities flatten out. 📊 In logistic regression, the natural log of odds (log-odds) forms the basis for linear prediction models.

 

  • Fractional (UK): e.g., 3/1 (“three to one”) means profit $3 per $1 stake
  • Decimal (Europe, AU): e.g., 2.00 means you get $2 back per $1 stake.
  • American/Moneyline: +200 means $200 profit on $100; –150 means bet $150 to win $100

 

  • Implied probability is derived from odds: e.g., decimal 2.00 means P = 1/2 = 50%
  • Over‑round is the sum of implied probabilities across all outcomes—usually exceeds 100% to ensure bookmaker profit
  • Gambler’s fallacy: Believing past events (e.g., multiple coin flips being heads) affect future probabilities
  • Ignoring the house edge: Taking odds at face value without recognizing embedded bookmaker profit margins
  • Confusion between formats: Misreading +200 as 200% probability rather than actual values.

Yes! Odds greater than 1 mean the probability is over 50%. For instance, odds of 3 (3:1) imply 75% probability Odds less than 1 indicate less likely events.

Discusson

  • Alex Morgan

    Alex Morgan

    1 year ago

    dfdsfsdfsd

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