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The probability of the number of defective circuit boards in a shipment is given in the tables for two different companies.
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).
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