What are the odds?
Why is it so silly to buy a lottery ticket? Why are some events considered less likely to happen than theothers? How do we make sense of the random nature of events around us?
We know that nothing is a hundred percent predictable- there is always some chance that things may not go as planned. For example, I buy some stocks or shares because they seem to be doing well. However, we cannot say with a hundred percent certainity that they willc ontinue to do so. At best, we can predict the most likely trend for our shares. Thus, by studying statistics, we find the “likelihood function ” for a given event to occur given the external parameters. Probability is the study of likelyhood of events happening. For example, a probability of 100% implies the event will always happen- if I roll a dice, the number will be between 1 and 6. A probability of 0% implies that the event cannot happen- I cannot pick a card reading 12 (of any suite) from a standard deck of cards. Further, the events may be mutually exclusive : If event A occurs, event B cannot, and vice versa. For example, in a coin toss, coin landing heads and coin landing tails are mutually exclusive. The events may be independent: event A does not effect event B in any way at all. There are three fundamental axioms of Probability, which require a conceptual understanding of set theory. 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.
While probability shows direct likelihood, odds express the ratio of success to failure.
From probability to odds (in favor):
\( \text{Odds} = \frac{P}{1 - P} \)
From odds to probability:
\( P = \frac{\text{Odds}}{1 + \text{Odds}} \)
Example: If \( P = 0.25 \), then
\( \text{Odds} = \frac{0.25}{0.75} = \tfrac{1}{3} \)
Conversely, if odds = 3, then
\( P = \frac{3}{1 + 3} = 0.75 \)
In Bernoulli trials with equally likely outcomes, odds are defined as
\( \text{Success} : \text{Failure} = S : F \)
and
This framework is useful for modeling coin tosses or dice rolls, where successes and failures are counted.
Odds are especially useful in betting, logistic regression, and risk analysis because they remain meaningful even at extreme probabilities (near 0 or 1), where probabilities flatten out. In logistic regression, predictions are made using the log of odds (log‑odds), which provides a linear scale suitable for modeling.
Yes! Odds greater than 1 indicate the event has a probability exceeding 50%. For instance, odds of 3:1 correspond to:
\( P = \frac{3}{1 + 3} = 0.75 \)
Odds less than 1 imply less likely events.
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