To determine which company should be chosen for purchasing circuit boards, we need to calculate the expected number of defective boards for both companies [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
The expected number of defective boards, [tex]\( E(X) \)[/tex], is calculated using the formula:
[tex]\[
E(X) = \sum_{i=0}^n x_i \cdot P(x_i)
\][/tex]
where [tex]\( x_i \)[/tex] represents the number of defective boards, and [tex]\( P(x_i) \)[/tex] represents the probability of having [tex]\( x_i \)[/tex] defective boards.
### Company A
The table of probabilities for Company A is:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline \# Defective (X) & 0 & 1 & 2 & 3 \\
\hline$P\left(x_i\right)$ & 0.21 & 0.30 & 0.41 & 0.08 \\
\hline
\end{tabular}
\][/tex]
Using the formula for the expected value:
[tex]\[
E(X_A) = (0 \cdot 0.21) + (1 \cdot 0.30) + (2 \cdot 0.41) + (3 \cdot 0.08)
\][/tex]
Breaking it down:
[tex]\[
E(X_A) = 0 + 0.30 + 0.82 + 0.24 = 1.36
\][/tex]
### Company B
The table of probabilities for Company B is:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline \# Defective (X) & 0 & 1 & 2 & 3 \\
\hline$P\left(x_i\right)$ & 0.18 & 0.32 & 0.28 & 0.22 \\
\hline
\end{tabular}
\][/tex]
Using the formula for the expected value:
[tex]\[
E(X_B) = (0 \cdot 0.18) + (1 \cdot 0.32) + (2 \cdot 0.28) + (3 \cdot 0.22)
\][/tex]
Breaking it down:
[tex]\[
E(X_B) = 0 + 0.32 + 0.56 + 0.66 = 1.54
\][/tex]
### Decision
The expected number of defective boards for Company A is 1.36 and for Company B is 1.54. Since a lower expected value of defective boards is preferable, we should choose the company with the lower expected number of defective boards.
Therefore, Company A should be chosen for purchasing boards as it has a lower expected number of defective boards (1.36) compared to Company B (1.54).
