The aim of this project is to examine the drop chances of blueprints from bosses with a particular emphasis on the following question: 'How many more runs until I get all my blueprints?!'.
We examine the following warframes: Rhino, Excalibur, Nyx, and Atlas (if you wish to see the behaviour of another warframe, feel free to email me). Though this project examines loot drop chances for warframes in particular, it is applicable to any other game with drop chances.
In all the plots, the green line is the probability of obtaining all warframe parts after X runs or less. This a cumulative probability plot and our main focus. The red line on the other hand is the probability of obtaining all the warframe parts after X runs exactly.
So for instance looking at Rhino's plot below (first plot), we have a ~0.02 chance of all Rhino parts dropping after exactly 10 runs (red line). However, we have a ~0.91 chance of getting all Rhino parts in any of the runs leading up to the 10th run (green line).
The graphs are obtained through numerical means rather than exact computation hence introducing some error though insignificant in this case. Crudely speaking, each plot below simulates 50 consecutive boss runs and repeats this process 100,000 times to average the probabilities, thus reducing errors stemming for numerical computation.
As for the correctness of the model, we stored the amount of times each item was dropped during the simulation. We then calculated the drop chance of each item purely from this data and the results were no more than 5% from the input probabilities fed into the simulation, thus confirming that simulation is functioning correctly. The input probabilities were obtained from the warframe wikia (see Credits section).
If attempting 10 runs, you have a ~90% chance of obtaining all parts (see below).
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If attempting 10 runs, you have a ~90% chance of obtaining all parts (see below).
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If attempting 10 runs, you have a ~90% chance of obtaining all parts (see below).
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If attempting 10 runs, you have a ~90% chance of obtaining all parts (see below).
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If attempting 29 runs, you have a ~90% chance of obtaining all parts of this
unfortunate Warframe (see below).
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Question: Since the probability of obtaining all parts of Atlas on the 10th and 11th run is roughly 90%, does this mean if I didn't obtain all parts on my 10th run, then my next run (11th run) has a probability of 90% of obtaining all parts?
Answer: No, that is false. This is known as the Gambler's Fallacy. For the 11th run, your probability of obtaining the last item (item 'c') will not be 90% but equal to the probability of that item dropping in exactly one instance run (in this case 22.6%). What about my 12th run? Assuming you are starting from your 10th run, then the probability of obtaining your last item is equal to the probability of that item dropping in exactly two instance runs or less (in this case 4*(22.6 * 38*7) = 35%). And so forth.
Now, if you have completed 10 runs and not all parts have dropped, then the next run will not have a 90% drop chance. But, if you start a new series of 10 runs from this point, then over the span of those 10 runs, you have a 90% chance of obtaining all parts. Actually, the new series of 10 runs will have a higher probability than the first series of 10 runs (i.e. higher than 90%) because you already have 2 out of 3 pieces.
Othman Alikhan, oz.alikhan@gmail.com
- Rhino probabilities: http://warframe.wikia.com/wiki/Nyx
- Excalibur probabilities: http://warframe.wikia.com/wiki/Excalibur
- Nyx probabilities: http://warframe.wikia.com/wiki/Nyx
- Atlas probabilities: http://warframe.wikia.com/wiki/Atlas