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On Monster Power 1, the drop rate for each key 10%. If I do 10 runs, the chance of having one key dropped in these 10 runs is 1-(1-10%)^10, or about 65%. After getting 3 keys, I can try to get a loot from a uber with a chance of 10%. I will need 3 loot from uber bosses to craft one ring. How do I calculate the Expected Value of total number of runs I will need?

Edit: from wikipedia, I just learned the Mean of a Geometric Distribution is 1/p. So the Expected Value of getting a key drop on Monster Power 1 should be 10 runs. But then, if 10 is the mean, why would 1-(1-10%)^10 ~= 65%?

user1032613
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  • Expected Value does not work like that. One should find the probability of a hellfire ring creation process turning successful, and go from there. Indeed, 1/p gives you in that case the number of runs to -almost- guarantee one ring, if you define, say, < 99% as "success". – DrFish Jun 18 '13 at 18:45
  • the (1-10%)^10 that you are using is chance *NOT* to get a key/organ in 10 runs. The 10% on MP1 means that there is 90% chance of NOT getting the key/organ. So in order to find the chance of getting a key/organ in N runs on X MP, you should use the formula that you showed: 1-(1-X*10%)^N with an exception when X = 0. So 1-(1-10%)^10 ~= 65% because there is a 65% chance of getting a key/organ in 10 runs. – Novarg Jun 18 '13 at 19:11
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    btw, if you have 3 key sets, you can easily find people willing to carry you on MP10 for a set of organs, so you don't really have to try for organ parts on mp1. – l I Jun 18 '13 at 19:35
  • The formula for expected value in the general case is Integral(0,n,P(x))=z, where P(x) is the probability function (.1), z is the target (1 key), and n is the expected value. Given that P(x) is a constant in this case, n=z/P() – Sconibulus Jun 19 '13 at 13:47

3 Answers3

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Your formula is correct: if the chances of getting a ring in one run are 1/10, then the expected number of runs is 10. And if that is the case, then the expected number of runs to get 3 rings is 3*10=30 runs.

Once you have all three keys, you craft them into a portal, which you use to get a chance at one of three organs. Assuming these organs also have a 10% drop rate, that means you'll need an average of 10 portals ( = 30 keys = 300 key-runs) per organ. Since you need 3 organs, your overall expected number of runs is 900 for all the keys you need, plus another 30 to get the organs, for a total of 930 (at MP1).

In general, if k is the probability of a key dropping and o is the probability of an organ dropping, your expected number of runs (including both keys and organs) will be 9/po + 3/o.

Note that the error in your logic is that you appear to be assuming the expected value is the same as the point at which you have 50% chance of succeeding - it's not. Take coin-flips for example; you have a 50% chance of getting heads after only one flip, but the expected number of flips to see a heads is 2.

BlueRaja - Danny Pflughoeft
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  • I'm confused about where the 65% come from... apparently the limit as p approaches zero, the chance of having a success in the first 1/p trails approaches to 1-(1/e), or about 63% - my 65% is obviously approaching that, but that value seems so random. How do I make sense of the 65% and the EV of 10 runs? – user1032613 Jun 18 '13 at 19:18
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    @user1032613: I hinted at that in my last paragraph, but a more in-depth answer is off-topic for this site. Please ask that on Math.SE and link to the question from here, and I will answer it there. – BlueRaja - Danny Pflughoeft Jun 18 '13 at 19:26
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    Thanks very much for following site topic rules :) Have an upvote! (I posted here: http://math.stackexchange.com/questions/423923/whats-the-expected-value-of-having-a-success-in-a-geometric-distribution) – user1032613 Jun 18 '13 at 20:29
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fbueckert's table still isn't quite right. Here's a fixed version:

+----+-------+----------+---------+--------+
| MP | 1 key | 1 portal | 1 organ | 1 ring |
+----+-------+----------+---------+--------+
|  0 |    20 |       60 |    1200 |   3600 |
|  1 |    10 |       30 |     300 |    900 |
|  2 |     5 |       15 |      75 |    225 |
|  3 |     4 |       10 |      34 |    100 |
|  4 |     3 |        8 |      19 |     57 |
|  5 |     2 |        6 |      12 |     36 |
|  6 |     2 |        5 |       9 |     25 |
|  7 |     2 |        5 |       7 |     19 |
|  8 |     2 |        4 |       5 |     15 |
|  9 |     2 |        4 |       4 |     12 |
| 10 |     1 |        3 |       3 |      9 |
+----+-------+----------+---------+--------+

It looks like MP3 and MP4 never got the previous fix, while MP6 and MP7 were rounded down instead of up (except in the first column). Here's my work if anyone wants to check it:

+----+-------+----------+---------+--------+
| MP | 1 key | 1 portal | 1 organ | 1 ring |
+----+-------+----------+---------+--------+
|  0 | 20    | 60       | 1200    | 3600   |
|  1 | 10    | 30       | 300     | 900    |
|  2 | 5     | 15       | 75      | 225    |
|  3 | 10/3  | 10       | 100/3   | 100    |
|  4 | 5/2   | 15/2     | 75/4    | 225/4  |
|  5 | 2     | 6        | 12      | 36     |
|  6 | 5/3   | 5        | 25/3    | 25     |
|  7 | 10/7  | 30/7     | 300/49  | 900/49 |
|  8 | 5/4   | 15/4     | 75/16   | 225/16 |
|  9 | 10/9  | 10/3     | 100/27  | 100/9  |
| 10 | 1     | 3        | 3       | 9      |
+----+-------+----------+---------+--------+

Sorry about answering instead of commenting. Apparently I need reputation to comment but not to answer. The MP6/7 differences are minor, but the MP3/4 differences are significant.

Brythan
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1

As everyone pointed out, there is no way to guarantee a specific key drop. It gets pretty close, but it never quite gets there.

For expected value, it's just adding the chances. 10% chance of a drop means the expected value is 10. 20% means you the expected value is 5, etc.

Monster Power        1 key   1 portal   1 organ    1 ring
---------------------------------------------------------
No Monster Power (5%):  20         60     1,200     3,600
MP 1                    10         30       300       900
MP 2                     5         15        75       225
MP 3                     4         12        48       144
MP 4                     3          9        27        81
MP 5                     2          6        12        36
MP 6                     2          5         8        24
MP 7                     2          4         6        18
MP 8                     2          4         5        15
MP 9                     2          4         4        12
MP 10                    1          3         3         9

This makes the assumption that you do all your runs at the exact same MP. I rounded up to the nearest whole value.

Frank
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    what threshold are you using for "guarantee"? – Sconibulus Jun 18 '13 at 18:31
  • 100%, as rounded to two decimal places. Without that, all I'd get is very long decimals in the 99% range. – Frank Jun 18 '13 at 18:35
  • 100%, though, is much different from expected value. – murgatroid99 Jun 18 '13 at 18:38
  • I guess this does not account for the same key type being dropped. The game does not guarantee distinct key drops. So, in the worst case, you will need SEVEN key runs for ONE organ. – DrFish Jun 18 '13 at 18:41
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    -1 there is no number of runs that will "guarantee" a drop of anything. Since the chances of dropping a key in one run is 1/10 at MP1, the expected number of runs for one key is 10. Where are you getting "94" from!? – BlueRaja - Danny Pflughoeft Jun 18 '13 at 18:43
  • Thanks for your effort, but I didn't post here to find someone to plug in my (possibly incorrect) formula onto a spreadsheet. I'm looking for a correct expression w.r.t probability p, and hopefully an explanation of how you derived the expression. – user1032613 Jun 18 '13 at 18:46
  • Bugger.. After 10 hours of coding, I knew I was doing something weird. Undo! Undo! – DrFish Jun 18 '13 at 18:52
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    @fbueckert Okay, I think you got it correct assuming the EV is 1/p. Although I don't think the "EV stays constant" from MP5-MP10 is true. Rounding errors? – user1032613 Jun 18 '13 at 19:09
  • @user1032613 There's no way to cut a run short. From MP5-9, you need to do two runs per key to get at or above the expected value. I could add in decimals if you want, but that's straying towards pure math, and not taking game mechanics into account. – Frank Jun 18 '13 at 20:53
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    @fbueckert although it's true that you can't do a half run, doing 36 runs on MP5 and on MP10 definite will give different chances of getting keys. Nothing is guaranteed to drop no matter how much you play. And no thanks, I don't want you to add in decimals just to make me happy. StackEx is for discussions; I'm not paying you money to boss you around. Thanks for all the help tho =) – user1032613 Jun 18 '13 at 22:28
  • You cannot round down before doing the multiplication. Doing the rounding first leads to rounding errors which, as @user1032613 points out, are obviously incorrect. – BlueRaja - Danny Pflughoeft Jun 19 '13 at 15:33
  • @BlueRaja-DannyPflughoeft Pray tell, how can you do 1.6 runs? This is where math conflicts with game mechanics. For pure math accuracy, yes, you can add in decimals. You can't get a key (or an organ) halfway through a run, though. That's a mechanical limitation. – Frank Jun 19 '13 at 15:39
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    It is not "pure math" vs. impure math, it is simply correct vs. incorrect. You cannot do 1.6 runs any more than the average family can have 1.6 children; but you cannot round that up/down in the middle of a calculation, or you will find that the expected number of total children in 10 families is 10 or 20, rather than the correct answer of 16. In the same way, by rounding before multiplying you find that the expected number of runs to get a ring is the same for MP5 and MP9, which is obviously incorrect. – BlueRaja - Danny Pflughoeft Jun 19 '13 at 15:59
  • Ah, I get what you're saying. Yeah, I didn't take that into account. I'll resolve that. – Frank Jun 19 '13 at 16:05
  • That looks much better, I'll remove my -1. Just so I can verify the numbers - what are the percentages for key/organ drops in MP1 thru MP10 (where do you find them?) – BlueRaja - Danny Pflughoeft Jun 19 '13 at 16:33
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    @BlueRaja-DannyPflughoeft It's 10% per MP. http://us.battle.net/d3/en/blog/7597724/New_Event_The_Infernal_Machine-10_15_2012 - Check the Components section. – Frank Jun 19 '13 at 16:43