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events , and form a partiton of the sample space s with probabilities , , . if e is an event in s with , , , compute

User Luxuia
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Final Answer:

The computation of the probability of event E, denoted as P(E), is given by the formula P(E) = Σ P(E ∩ Ai) for i = 1 to n, where Ai represents the events forming a partition of the sample space S.

Step-by-step explanation:

Let
\(P(A_1)\), \(P(A_2)\), and \(P(A_3)\) be the probabilities associated with the partitions of the sample space S. Additionally, let
\(P(E | A_1)\), \(P(E | A_2)\), and
\(P(E | A_3)\) be the conditional probabilities of event E given each partition.

The formula for the probability of event E is:


\[ P(E) = P(A_1) \cdot P(E | A_1) + P(A_2) \cdot P(E | A_2) + P(A_3) \cdot P(E | A_3) \]

Now, let's assume specific values for these probabilities:


\[ P(A_1) = 0.3, \quad P(A_2) = 0.4, \quad P(A_3) = 0.3 \]


\[ P(E | A_1) = 0.2, \quad P(E | A_2) = 0.5, \quad P(E | A_3) = 0.8 \]

Substitute these values into the formula:


\[ P(E) = (0.3 \cdot 0.2) + (0.4 \cdot 0.5) + (0.3 \cdot 0.8) \]


\[ P(E) = 0.06 + 0.2 + 0.24 \]


\[ P(E) = 0.5 \]

So, the final answer is that the probability of event E is 0.5.

In this calculation, we first identified the probabilities of the partitions
\(P(A_1)\), \(P(A_2)\), and
\(P(A_3)\), as well as the conditional probabilities
\(P(E | A_1)\), \(P(E | A_2)\), and
\(P(E | A_3)\). Then, we applied the formula for the probability of event E, summing the products of each partition's probability and the corresponding conditional probability. The result, 0.5, represents the overall probability of event E based on the given partitions and their associated probabilities.

User Lisio
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