Events, conditional probability and Bayes' theorem
In Class 11, probability is built on sets. The sample space S lists every outcome of an experiment, and an event is a subset of S. For any events A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive they cannot happen together, so P(A ∩ B) = 0 and their probabilities simply add. The complement A′ contains every outcome not in A, and P(A′) = 1 − P(A), which is often the quickest route to questions that ask for the chance of at least one success.
Conditional probability P(A | B) is the probability of A given that B has already happened, equal to P(A ∩ B) ÷ P(B). Events are independent when one does not affect the other, which means P(A ∩ B) = P(A) × P(B); for two coin tosses, the chance of two heads is ½ × ½ = ¼. The total probability rule and Bayes' theorem let you reverse a condition, for example finding the probability that a defective item came from a particular machine. In Class 12, random variables and the binomial distribution describe repeated independent trials.
Why it matters
Probability is used to set insurance premiums, interpret medical test results, forecast weather, control quality in factories and design games and simulations. Bayes' theorem in particular shows how new evidence should change a belief, an idea used in spam filters and machine learning. The topic carries significant weight in Class 12 board papers and entrance tests.
Key terms
- Sample space
- The set of all possible outcomes of a random experiment.
- Mutually exclusive events
- Events that cannot occur at the same time in a single trial.
- Independent events
- Events where the occurrence of one does not change the probability of the other.
- Conditional probability
- The probability of one event given that another event has already occurred.