An inspector at a manufacturing plant examines boxes containing table lamps that are ready for shipment. A box may be intact, slightly damaged, or badly damaged. $90\%$ of all boxes are intact, $6\%$ are slightly damaged, and $4\%$ are badly damaged. If a box shows signs of damage, the inspector may open the box and examine the lamp within. The lamp will be found to be either unbroken or broken. If the lamp is broken, it is discarded, and the loss of the lamp costs the company $$10.$ If it is unbroken, it is repackaged at a cost of $$1,$ then shipped.

Experience has shown that the lamp in a slightly damaged box has an $80\%$ chance of being unbroken, but the lamp in a badly damaged box has only a $40\%$ chance of being unbroken. $($Lamps in undamaged boxes are always unbroken.$)$ If a broken lamp is shipped to a customer, it generates $$20$ in costs to the company, due to replacement costs and customer dissatisfaction. Ignore all costs other than those described in this problem. The inspector will adopt an inspection strategy that will minimize the expected cost to the company.

Following the optimal strategy will result in an average cost of $42\mathrm{.}4$ cents per box. The EVPI for this problem is $6\mathrm{.}4$ cents per box. From this it follows that

Group of answer choices

having perfect information in this problem would reduce average cost to $6\mathrm{.}4$ cents per box.

if the inspector could know whether a box was intact, slightly damaged, or badly damaged without looking at it$,$ it would reduce the average cost per box to $36$ cents.

if the inspector knew whether the lamp in a box was broken without opening the box, this would reduce the average cost to $36$ cents per box.

if the inspector opened every box, it would increase average costs by $6\mathrm{.}4$ cents per box.

if the inspector had to decide whether to open a box before seeing the condition of the box, the average cost per box would jump to $48\mathrm{.}8$ cents.