Chapter 6, Problem 87CR

A company uses three different assembly lines—A₁, A₂ and A₃—to manufacture a particular component. Of those manufactured by A₁, 5% need rework to remedy a defect, whereas 8% of A₂’s components and 10% of A₃’s components need rework. Suppose that 50% of all components are produced by A₁, 30% are produced by A₂ and 20% are produced by A₃.

1. a. Construct a tree diagram with first-generation branches corresponding to the three lines. Leading from each branch, draw one branch for rework (R) and another for no rework (N). Then enter appropriate probabilities on the branches.
2. b. What is the probability that a randomly selected component came from A₁ and needed rework?
3. c. What is the probability that a randomly selected component needed rework?

To determine

Draw a tree diagram with first-generation branches corresponds to the three lines.

Answer to Problem 87CR

The tree diagram with first-generation branches corresponds to the three lines is:

Explanation of Solution

Calculation:

The given information is that about 50% of component manufactured by $A_1$ among that 5% need rework to remedy a defect, about 30% of component manufactured by $A_2$ among that 8% need rework to remedy a defect and about 20% of component manufactured by $A_3$ among that 10% need rework to remedy a defect.

Here, the probability of component manufactured by $A_1$ is 0.5.

The probability of component manufactured by $A_2$ is 0.3.

The probability of component manufactured by $A_3$ is 0.2.

The probability of component manufactured by $A_1$, need rework to remedy a defect is 0.05.

The probability of component manufactured by $A_2$, need rework to remedy a defect is 0.08.

The probability of component manufactured by $A_3$, need rework to remedy a defect is 0.1.

Tree diagram:

Tree diagram is a graphical representation of sample points. From a tree diagram, one can easily identify the sample points and number of sample points in the sample space.

The following tree diagram represents the given situation:

To determine

Compute the probability that a randomly selected component came from $A_1$ and needed rework.

Answer to Problem 87CR

The probability that a randomly selected component came from $A_1$ and needed rework is 0.025.

Explanation of Solution

Calculation:

The probability that a randomly selected component came from $A_1$ and needed rework is calculated as follows:

$\begin{array}{c}P\left(A_1\cap R\right)=\left(0.5\right)\times \left(0.05\right)\\ =0.025\end{array}$

Thus, the probability that a randomly selected component came from $A_1$ and needed rework is 0.025.

To determine

Find the probability that a randomly selected component need rework.

Answer to Problem 87CR

The probability that a randomly selected component need rework is 0.069.

Explanation of Solution

Calculation:

The probability that a randomly selected component need rework is calculated as follows:

$\begin{array}{c}P\left(R\right)=\left(0.5\right)\left(0.05\right)+\left(0.3\right)\left(0.08\right)+\left(0.2\right)\left(0.1\right)\\ =0.025+0.024+0.02\\ =0.069\end{array}$

Thus, the probability that a randomly selected component need rework is 0.069.