Question

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

Answer 1

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.