5. An assembly plant receives its voltage regulators from three different suppliera, 60% from supplier B1, 30% from supplier B2, and 10% from supplier B3. Also 95% of the voltage regulators from B1, 80% from B2, and 65% of from B3 perform according to specifications.
Let assume A = according to specifications.
(5 points)
(a) Write down the given values of 60%, 30%, 10%, 95%, 80% and 65% in terms of events A, B1, B2, B3, and operations union, intersection, complement and condition.
What is the probability that:
(b) any one voltage regulator received by the plant will perform according to specifications?
Write down the formula you are using in terms of A and B's.
© a particular voltage regulator, which is known to perform according to specifications, came from supplier B3? Write down the formula you are using in terms of A and

### (a) Defining the Given Values in Terms of Events

Let’s define the events properly:
- \( B_1 \): The event that a voltage regulator comes from supplier \( B_1 \).
- \( B_2 \): The event that a voltage regulator comes from supplier \( B_2 \).
- \( B_3 \): The event that a voltage regulator comes from supplier \( B_3 \).
- \( A \): The event that a voltage regulator performs according to specifications.

We are given:
- \( P(B_1) = 0.60 \) (60% from supplier \( B_1 \))
- \( P(B_2) = 0.30 \) (30% from supplier \( B_2 \))
- \( P(B_3) = 0.10 \) (10% from supplier \( B_3 \))

Also, the conditional probabilities that the voltage regulators perform according to specifications given that they come from each supplier:
- \( P(A | B_1) = 0.95 \) (95% of regulators from \( B_1 \) perform to specification)
- \( P(A | B_2) = 0.80 \) (80% of regulators from \( B_2 \) perform to specification)
- \( P(A | B_3) = 0.65 \) (65% of regulators from \( B_3 \) perform to specification)

### (b) Probability That Any One Voltage Regulator Performs According to Specifications

The probability that any voltage regulator performs according to specifications, \( P(A) \), can be found using the law of total probability:

\[
P(A) = P(A \cap B_1) + P(A \cap B_2) + P(A \cap B_3)
\]

Using the definition of conditional probability \( P(A \cap B_i) = P(A | B_i) \cdot P(B_i) \), we get:

\[
P(A) = P(A | B_1) \cdot P(B_1) + P(A | B_2) \cdot P(B_2) + P(A | B_3) \cdot P(B_3)
\]

Substitute the values:

\[
P(A) = (0.95 \cdot 0.60) + (0.80 \cdot 0.30) + (0.65 \cdot 0.10)
\]

Now, calculate:

\[
P(A) = 0.57 + 0.24 + 0.065 = 0.875
\]

Thus, the probability that a voltage regulator performs according to specifications is:

\[
P(A) = 0.875
\]

### (c) Probability That a Regulator Came from \( B_3 \), Given That It Performs According to Specifications

We need to find \( P(B_3 | A) \), the probability that a regulator came from supplier \( B_3 \) given that it performs according to specifications. By Bayes' Theorem, we have:

\[
P(B_3 | A) = \frac{P(A \cap B_3)}{P(A)}
\]

We already know that \( P(A \cap B_3) = P(A | B_3) \cdot P(B_3) \), so:

\[
P(B_3 | A) = \frac{P(A | B_3) \cdot P(B_3)}{P(A)}
\]

Substitute the values:

\[
P(B_3 | A) = \frac{0.65 \cdot 0.10}{0.875} = \frac{0.065}{0.875} \approx 0.0743
\]

Thus, the probability that a voltage regulator came from supplier \( B_3 \), given that it performs according to specifications, is approximately:

\[
P(B_3 | A) \approx 0.0743
\]

### Summary of Results:
- \( P(A) = 0.875

An assembly plant receives its voltage regulators from three different suppliers: 60% from supplier B1, 30% from supplier B2, and 10% from supplier B3. Also, 95% of the voltage regulators from B1, 80% from B2, and 65% from B3 perform according to specifications. (a) Write down the given values of 60%, 30%, 10%, 95%, 80%, and 65% in terms of events A, B1, B2, B3, and operations like union, intersection, complement, and condition. (b) What is the probability that any one voltage regulator received by the plant will perform according to specifications? (c) What is the probability that a particular voltage regulator, known to perform according to specifications, came from supplier B3?

Explore how to calculate the probability that a voltage regulator will perform according to specifications, and the likelihood that a regulator came from supplier B3 using conditional probability and Bayes' Theorem. This problem covers probability concepts at a college level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation