$*$PLEASE SHOW ME THE STEP$-$BY$-$STEP SOLUTIONS AS I AM LOOKING FORWARD GUIDANCE ON WHAT I AM CURRENTLY DOING WRONG! THANK YOU: A manufacturing company produced items that have a probability y of being defective. These items are produced in stacks of $150.$ Past experience indicated that p for each item in a stack either $0\mathrm{.}05$ or $0\mathrm{.}25.$ Furthermore, in $80\%$ of the stacks produced, p equals $0\mathrm{.}05($so p equals $0\mathrm{.}2520\%$ of the stacks$).$ These items are then used in an assembly, and ultimately their quality is determined before the final assembly leaves the factory. Initially the company

Can either screen each item in a stack at a cost of $$10$ per item and replace defective items or use the items directly without screening. If the latter action is taken, the cost of rework is ultimately $$100$ per defective item. Because screening required scheduling of inspectors and equipment, the decision to screen or not screen must be made two days before the screening is to take place. However, one item can be taken from the stack and sent to a laboratory for inspection, after which its quality $($defective or non$-$defective$)$ can be reported before the screen$/$no$-$screen decision must be made. The cost of this initial inspection is $$125.($Note: round up to three decimal places if necessary$)$

a$)$ A single item is inspected in advance. Use the conditional and prior probabilities to calculate the posterior$/$revised probabilities of the respective states of nature for each of the two possible outcomes of this inspection. Show your work$/$calculations$.(14$ points$)$

b$)$ Draw and solve the decision tree for this entire problem to determine the optimal decision strategy that the manufacturing company should take $($show your work$/$calculations$).$ Verbally communicate the decision strategy. $(21$ points$)$

c$)$ Find the expected value of sample information. If the cost of using the laboratory to inspect the single item in advance is open to negotiation, how large can the cost of using the laboratory be and still be worthwhile $($i$.$e$.,$ what is the maximum amount that the manufacturing company is willing to pay for the pre$-$screening lab inspection$)?(3$ points$)$