Consider 3 urns. Urn A contains 2 white and 4 red balls; urn...

Consider 3 urns. Urn $A$ contains 2 white and 4 red balls; urn $B$ contains 8 white and 4 red balls; and um $C$ contains 1 white and 3 red balls. If 1 ball is selected from each urn, what is the probability that the ball chosen from urn $A$ was white, given that exactly 2 white balls were selected?

Key Concepts

Independent Events

Independent events are events whose occurrences do not affect each other's probability. When dealing with independent events, the probability of their joint occurrence is the product of their individual probabilities. This concept simplifies the analysis and computation in many problems involving random processes and selections.

Law of Total Probability

The Law of Total Probability is used to compute the probability of an outcome by considering all distinct ways in which that outcome can occur. It involves partitioning the sample space into mutually exclusive events and summing the probabilities across these channels, thereby connecting individual conditional probabilities to a comprehensive overall probability.

Conditional Probability

Conditional probability focuses on determining the probability of an event occurring given that another event has already occurred. It is typically expressed as P(A|B), which quantifies the likelihood of event A happening under the condition that event B has occurred, forming the basis for many decision-making and inference models in probability theory.

Bayes' Theorem

Bayes' Theorem provides a mathematical framework to update the probability of a hypothesis, given new evidence. It relates the conditional and marginal probabilities of stochastic events and is key to inferential statistics, allowing us to reverse conditionality — predicting the likelihood of the cause from the manifestation of an effect.

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