What are the odds?
Why is it so silly to buy a lottery ticket? Why are some events considered less likely to happen than the others? 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 will continue 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.
Probability measures how likely an event is to occur. It’s calculated as the number of favorable outcomes divided by the total number of possible outcomes. The result ranges from 0 (impossible event) to 1 (certain event)
Use this formula:
\( \text{Probability} = (\text{Number of favorable outcomes}) / (\text{Total outcomes}) \)
e.g., the chance of drawing a red marble from a bag of 3 red, 4 white, and 6 blue marbles is 7/13 if red or white counts as favorable
Addition Rule (OR): For mutually exclusive events, \( P(A or B) = P(A) + P(B) \).
Multiplication Rule (AND): For independent events, \( P(A and B) = P(A) × P(B) \)
According to the law of large numbers, as the number of trials increases, the experimental probability approaches the theoretical probability. So, flipping a fair coin thousands of times will yield close to a 50% heads result
A Bernoulli trial is an experiment with exactly two possible outcomes: success (e.g. “heads”) or failure (e.g. “tails”), each with a constant probability. These trials lay the foundation for binomial experiments like multiple coin flips
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