Use Bayes' theorem or a tree diagram to calculate the...

Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round all answers to four decimal places.
$P(A \mid B)=.8, P(B)=.2, P\left(A \mid B^{\prime}\right)=.3 .$ Find $P(B \mid A)$.

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Key Concepts

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It serves as the foundation for more complex probability calculations and models. Understanding conditional probability is key to analyzing situations where the likelihood of outcomes depends on certain conditions or prior events.

Bayes' Theorem

Bayes’ Theorem is a fundamental concept in probability theory that relates the conditional probabilities of events. It provides a way to update prior beliefs based on new evidence by expressing the probability of an event given another event in terms of the reverse conditional probability. This theorem is widely applied in various fields such as medicine, finance, and machine learning for decision-making under uncertainty.

Law of Total Probability

The Law of Total Probability is a rule that helps to compute the probability of an event by considering all possible mutually exclusive scenarios that lead to that event. It involves summing over the weighted probabilities of the event occurring under each condition. This law is essential when breaking down complex probability problems into simpler, more manageable parts.

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