Events D1, D2, and D3 are mutually exclusive and exhaustive...

Events $D_1, D_2$, and $D_3$ are mutually exclusive and exhaustive causes of two effects, $C$ and $C$. The probabilities of the causes are $P\left(D_1\right)=0.69, P\left(D_2\right)=0.05$, and $P\left(D_3\right)=0.26$. Given that $P\left(C \mid D_1\right)=0.10$, $P\left(C D_2\right)=0.36$, and $P\left(C D_3\right)=0.54$, what is the probability of effect $C^{\prime}$ ?

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Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has occurred. It forms the basis for updating probabilities based on new information and is central to understanding dependent events in probability theory.

Law of Total Probability

The law of total probability allows the computation of the overall probability of an event by partitioning the sample space into mutually exclusive events. By summing the probabilities of the event under each partition, weighted by the probability of each partition, one can determine the unconditional probability of that event.

Complementary Events

Complementary events are a pair of outcomes where one event represents all outcomes not covered by the other, ensuring that the probabilities of an event and its complement sum to one. This concept is frequently used to simplify probability calculations by subtracting the probability of an event from one.

Mutually Exclusive and Exhaustive Events

Mutually exclusive events are those that cannot happen simultaneously, and when a set of events is exhaustive, they cover all possible outcomes in the sample space. This property is crucial for methods like the law of total probability, as it ensures that exactly one of the events occurs in any trial.

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