For the information provided in Problem 9.64, what is the...

For the information provided in Problem 9.64, what is the probability that effect $C^{\prime}$ is caused by $D_1, D_2$, or $D_3$ ? In other words, what are:
(a) $P\left(D_1 \mid C^{\prime}\right)$,
(b) $P\left(D_2 \mid C^n\right)$.
(c) $P\left(D_3 \mid C^{\prime}\right)$ ?

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Key Concepts

Bayes' Theorem

Bayes' Theorem is a formula that relates the conditional and marginal probabilities of random events. It is used to update the probability estimate for a hypothesis as more evidence or information becomes available. In this context, Bayes' Theorem is the key tool for calculating the likelihood that each potential cause is responsible for the observed effect.

Law of Total Probability

The Law of Total Probability provides a way to break down complex probability calculations by considering all possible ways an event can occur. It is used to compute the overall probability of the observed effect by summing the contributions from all potential causes. This law is essential for correctly normalizing the probabilities when applying Bayes' Theorem, ensuring that the probabilities of all possible causes add up to one.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another related event has already taken place. It is a fundamental concept in probability theory that allows us to update the probability of an outcome based on new or additional information, which in this context involves understanding the probability of a specific cause given the occurrence of an effect.

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