EX 2.35 batch pass inspection

• Each day, Monday through Friday, a batch of components sent by a first supplier arrives at a certain inspection facility.
• Two days a week, a batch also arrives from a second supplier.
  • $80\%$ of all supplier 1’s batches pass inspection,
  • $90\%$ of supplier 2’s do likewise.

Question

What is the probability that, on a randomly selected day, two batches pass inspection?

We will answer this assuming that on days when two batches are tested,

• whether the first batch passes is independent of whether the second batch does so. Figure 2.13 displays the relevant information.

Figure 2.13 Tree diagram for Example 2.35

$$\begin{array}{rl}P\left(\text{two pass}\right)& =P\left(\text{two received}\cap \text{both pass}\right)\\ & =P\left(\text{both pass}\mid \text{two received}\right)\cdot P\left(\text{two received}\right)\\ & =\left[\left(.8\right)\left(.9\right)\right]\left(.4\right)=.288\end{array}$$