QUESTION -

We recently discovered a "minor" design flaw with QBUS Robots. About 2% of the robots are overly susceptible to sonic interference at a particular harmonic frequency to the extent that the robot can explode. We can identify the robots at risk by exposing them to a "sonic screwdriver". If this results in a reversal of the polarity of the neutron flow, then the robot is at risk.

If the robot is at risk, we can leave it as it is or try to fix it. There are two options for fixing it. We can replace capacitor A53 which resides behind the robot's eye stalk, or we can replace capacitor Q97 which sits above the hover-pad motor (it is these two capacitors whose frequencies may too closely match). Interestingly, both of these options carry their own risk of catastrophe.

QBUS Robots have defensive capabilities. During each operation there is a 1/50,000 chance that a static charge will awake the robot momentarily. During this time there is an 85% chance that the robot will identify the operation as a potential security risk. If it does, then it will perform a complete power up to investigate and address the security risk.

For the eyestalk operation, the loss of sight will reduce the robot's ability to identify the technician as an authorised entity - it will be unable to perform a visual authentication. In addition, if the technician becomes fearful, his voice may alter to the extent that he will fail the voice recognition authentication and will also be unable to state his 64 digit emergency code within the 60 second time window. Altogether, there is a 1% chance of him failing to authenticate. In this case, the robot will attempt to set-off an internal explosive and self-destruct entirely (with a 3 second warning). The explosive fails to explode three times in a thousand.

Additionally, if the robot begins its power up sequence after the eyestalk operation has begun, then the technician can try to access a tiny diagnostic button that would result in a manual shutdown. He would need to press the button before the robot completely powered up - that is, within about 15 seconds. The chance of him reacting quickly enough and attempting this is estimated to be 20%. If he attempts it, but presses the adjacent button by mistake then after the robot completes its power up, it will immediately set off its explosive charge (following the 3 second warning). There is a 15% chance of the technician making this mistake.

For the hover-pad operation, there is only a 1 in 10,000 chance of failing to authenticate. However, after a power-up, the robot will attempt to move out of the reach of the technician. Operations are always performed in a flat area with the robot constrained. Nevertheless, it is possible that under sufficient force from the robot and with the loss of fine movement control resulting from the removal of capacitor Q97, the robot will capsize. There is perhaps a 1 in 200 chance of this happening before the authentication process completes. If this happens then there is a 1:1 chance of the robot deciding it is under attack and initiating its self-destruction. If it capsizes and then decides that it is not under attack then it no longer needs to authenticate as it has established that is it 'safe'. Note that the diagnostic button is not accessible to the technician from under the hover-pad mount.

Assume these are the complete set of risks that would lead to a catastrophe. Further assume that the technicians choose randomly (by tossing a fair coin) as to whether they replace the A53 or the Q97. Finally, assume all robots will eventually be exposed to the dangerous sonic frequency.

Part A -

1. From the point at which the decision is made to try and fix a faulty robot, draw the probability tree for calculating the probability of a catastrophe. Pay careful attention to the format of the tree.

2. Solve the tree (showing your working on the probability tree) to determine the chance of a catastrophe.

Part B -

We are evaluating whether we should advise our technicians to never press the diagnostic button when doing the eye-stalk operation. Reverse the conditional probabilities to work out the chance that the engineer was the cause of a robot exploding while being repaired by either approach. That is, determine P(manual shutdown attempted | robot exploded)

Based on this information, should we advise our technicians to never press the diagnostic button when doing the eye-stalk operation? Why?

Part C -

QBUS Industries is an intelligent organisation and so would not randomly choose which capacitor to replace. Based on minimising the chance of the catastrophe, which chip should be replaced, and what is the new overall chance of a catastrophe occurring? Show your working.

Part D -

QBUS Industries is also a highly profitable organisation and so would actually use financial considerations for deciding which chip to replace.

Each Robot costs $200,000 to build. If a robot explodes in the lab, there would typically be no additional damage. If the robot exploded while doing its normal duties, there would be additional damage, estimated to be a further $500,000 on average. Replacing the chip behind the hover-pad is marginally more expensive, as the robot needs to be held rotated. It costs $6100 for the hover-pad operation compared with $6000 for the eye-stalk operation.

One Engineer suggested simply exposing all the robots to high levels of the dangerous sonic frequency in the safe condition of the lab. If they exploded, it would save money as 2% of $200,000 is less than the costs of repairs.

Draw and solve the decision tree for this problem, paying careful attention to formatting. Include the options of doing nothing and the Engineer's suggestion. Identify the decisions that you should make at each stage and the expected costs per robot across the whole robot fleet. You should utilise information you have already calculated in part 1 of the question, without the need to repeat it.