Game Items

Solution

We want to maximize the number of verified real items produced per unit time. First determine the average time to produce an item. Every 10 minutes there is a 10% chance to produce one item, so the expected time to produce one item is \[ \frac{10}{0.1} = 100 \text{ minutes}. \] Once an item is produced: \[ P(\text{real}) = 0.1, \quad P(\text{fake}) = 0.9. \] Testing takes 30 minutes. If an item is real: \[ P(\text{pass}) = 0.25, \quad P(\text{fail}) = 0.75. \] If an item is fake: \[ P(\text{fail}) = 1. \] Suppose we decide to test an item up to \(N\) times. If it passes we keep it, otherwise after \(N\) failures we discard it. A fake item can never pass, so success only occurs if the item is real and at least one test passes. The probability that a real item fails \(N\) times is \[ 0.75^N. \] So the probability a real item passes within \(N\) tests is \[ 1 - 0.75^N. \] Since only 10% of items are real, \[ P(\text{verified}) = 0.1(1 - 0.75^N). \] If the item is fake, it always fails and we perform all \(N\) tests. If the item is real, we stop at the first pass. The probability we reach test \(k\) is that the first \(k-1\) tests failed: \[ 0.75^{k-1}. \] So the expected number of tests for a real item is \[ 1 + 0.75 + 0.75^2 + \dots + 0.75^{N-1}. \] This geometric sum equals \[ \frac{1 - 0.75^N}{1 - 0.75} = 4(1 - 0.75^N). \] Now combine real and fake cases: \[ E[\text{tests}] = 0.9N + 0.1 \cdot 4(1 - 0.75^N). \] Each test takes 30 minutes, so testing time is \[ 30\left(0.9N + 0.4(1 - 0.75^N)\right). \] Adding the 100 minutes to produce the item gives \[ T_N = 100 + 30\left(0.9N + 0.4(1 - 0.75^N)\right). \] Verified real items per minute \[ R_N = \frac{0.1(1 - 0.75^N)} {100 + 30\left(0.9N + 0.4(1 - 0.75^N)\right)}. \] Now check several values. \[ N=1 \Rightarrow 0.000192 \] \[ N=2 \Rightarrow 0.000275 \] \[ N=3 \Rightarrow 0.000308 \] \[ N=4 \Rightarrow 0.000316 \] \[ N=5 \Rightarrow 0.000312 \] The rate peaks at \(N=4\).

Final answer: Test each item until it either passes or fails 4 times* If it has failed 4 tests, discard it and produce a new item. This strategy maximizes the number of verified real items per unit time. The resulting output is about 0.455 verified real items per day.