Sorry if this question is a bit lower-level, yet complicated, but I feel like there is something wrong and I cannot put my finger on it. This scenario is adapted from something I read elsewhere on the internet, but simplified to get the basic point across.
Lets say I have a system where I have different tiered shapes. Triangles can be bought from the store for 1 dollar. To make a square, you must combine two triangles. However, this combining has a 20% chance to be successful, in which case the 2 triangles combine to become 1 square. It has a 80% chance to fail, in which case the 2 triangles combine to form 1 failed triangle. What's the average cost for a square? You would buy 2 triangles at a time until you succeed. Then you can sell the excess failed triangles from failed attempts for 1 dollar each. However, you cannot combine the triangles from failed attempts to save money (lets just say they are failed and only can be combined once to make the situation simpler, but again, can be sold to the store for 1 dollar).
Person 1 says the following: Well since it is a 20% success rate, then on average I should have 5 combinations needed to get 1 square. 5 combinations costs 10 triangles, and I would be left with 1 square and 4 failed triangles. I bought 10 triangles for 10 dollars, but then I sold the 4 failed ones for 4 dollars, so the net cost is 6 dollars per square.
Person 2 says the following: Since this is a cumulative probability of independent events we can predict the probability of getting a square after $n$ combinations using $1-(1-x)^n$ where $x$ is the probability of success per event and $n$ is number of attempts. This handy formula is derived from multiplying the probability of failure after each event. We want the average number of times to combine, so we want to find the $n$ that corresponds to 0.5 probability (meaning that after $n$ combinations, there will be a 50% chance of getting at least 1 success. Since the success rate is 20%, we do $0.5=1-(1-0.2)^n$ and solve for $n$. We find $n\approx3.106$. This means we need on average about 3.106 combinations, or 6.212 triangles to be bought. Also, I can sell the failed triangles. Since all but one of the combinations fail, then if there are 3.106 combinations needed on average, then we would have 2.106 failed combinations, which can be sold for 2.106 dollars. We do 6.212-2.106 to get the net cost of 4.106 dollars per square.
Who is more correct? For the record I am pretty certain that person 1's interpretation of average needed combinations is naive and incorrect, but I am not sure if person 2's calculation for the number of failed triangles is correct. Even more confusing, other people have verified person 1's concept using computer programs running over millions of trials, finding that the result is always 6 dollars. Thus, person 1's interpretation is generally accepted in this community. This, again, seems impossible considering that the average needed combination calculation seems to be naive and incorrect.
And for fun, here's the full scenario (still adapted): When combining a shape, there is a 50% chance to return a failed shape of the same tier, 20% to upgrade 1 tier, and 30% chance return a failed shape of 1 tier lower. If you combine 2 triangles, since triangles are the lowest tier, it has an 80% chance to give a failed triangle and 20% chance to give a square. If I were to combine two squares, there's a 50% chance to return a failed square, a 20% chance to return a pentagon, and a 30% chance to return a failed triangle. The same goes for combining pentagons to potentially get a hexagon. Find the average cost per square, assuming that triangles are worth 1 dollar. Then use that average cost to calculate the average cost per pentagon (failed squares can be sold for the average cost per square) and average cost per hexagon (failed pentagons can be sold for the average cost per pentagon, just like squares).
Thank you so much. I hope we can get to the bottom of this.