- The expected value for Company A is calculated as \(E[X_A] = (0 \times 0.21) + (1 \times 0.30) + (2 \times 0.41) + (3 \times 0.08) = \boxed{1.36}\).
- The expected value for Company B is calculated as \(E[X_B] = (0 \times 0.18) + (1 \times 0.32) + (2 \times 0.28) + (3 \times 0.22) = \boxed{1.54}\).
- Since \(1.36 < 1.54\), Company A should be chosen because it has a lower expected number of defective circuit boards.
- Therefore, the preferred company is \(\boxed{\text{Company A}}\).
Explanation
- Calculate the expected number of defective boards for Company A
To determine which company to choose, we calculate the expected number of defective circuit boards for both companies. The expected value \(E[X]\) of a discrete random variable \(X\) is calculated by multiplying each possible outcome \(x_i\) by its probability \(P(X=x_i)\) and then summing these products:
\[E[X] = \sum x_i \cdot P(X=x_i)\]
For Company A, the outcomes are 0, 1, 2, and 3 with probabilities 0.21, 0.30, 0.41, and 0.08, respectively.
\[E[X_A] = (0 \times 0.21) + (1 \times 0.30) + (2 \times 0.41) + (3 \times 0.08)\]
\[E[X_A] = 0 + 0.30 + 0.82 + 0.24 = 1.36\]
So, the expected number of defective boards for Company A is 1.36.
- Calculate the expected number of defective boards for Company B
For Company B, the outcomes are 0, 1, 2, and 3 with probabilities 0.18, 0.32, 0.28, and 0.22, respectively.
\[E[X_B] = (0 \times 0.18) + (1 \times 0.32) + (2 \times 0.28) + (3 \times 0.22)\]
\[E[X_B] = 0 + 0.32 + 0.56 + 0.66 = 1.54\]
So, the expected number of defective boards for Company B is 1.54.
- Compare and conclude
Comparing the two expected values, \(E[X_A] = 1.36\) and \(E[X_B] = 1.54\), we see that Company A has a lower expected number of defective circuit boards. Since we want to minimize the number of defective items, we should choose Company A.
Examples
In supply chain management, calculating the expected number of defective units helps companies select vendors that provide higher quality products. By comparing the expected values, businesses can make data-driven decisions to minimize waste and improve overall efficiency. For instance, if you are buying 1,000 boards, choosing Company A instead of Company B would save you, on average, \(1000 \times (1.54 - 1.36) = 180\) defective boards.