This was posted on the steam forums:
Been lots of talk about math here. Here's the actual numbers.
These calculations presume no bugs and we take what they say at face value. (Which I know to be false - myself and many others haven't gotten a SINGLE drop, new or duplicate - but that aside...) Valve says a drop is expected, on average, every 38 minutes. There are a total of 18 unlocks.
Here's a walkthrough of the math, and ALL numbers for those curious.
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TIME TO GET UNIQUE ITEM #1 (HAVE 0 UNIQUE ITEMS, MISSING 18 UNIQUE ITEMS)
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Let's say you start with an empty backpack. You have no items at all.
Probability that the first unlock - after 38 minutes - is 100%, since all are new.
So average time to get FIRST item is 38 minutes. That's our starting point.
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TIME TO GET UNIQUE ITEM #2 (HAVE 1 UNIQUE ITEMS, MISSING 17 UNIQUE ITEMS)
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At this point, you have 1 item, and you're looking for your second.
* The probability of it taking ONE try to get a new unlock is 17/18.
* The probability of it taking TWO tries to get a new unlock is 17/18 * 1/18 (the probability it's good, times the probability that the first was bad)
* The probability of it taking THREE tries to get a new unlock is 17/18 * 1/18 * 1/18 (the probability it's good, times the probability that the first and second were both bad)
Put mathematically, the average time to get the second unlock is:
SUM(1...infinity) 38 * 17/18 * (1/18)^(n-1)
Numbers come out to:
Probability of taking 1 tries, 38 minutes, is 0.94444444
Probability of taking 2 tries, 76 minutes, is 0.05246914
Probability of taking 3 tries, 114 minutes, is 0.00291495
Probability of taking 4 tries, 152 minutes, is 0.00016194
Probability of taking 5 tries, 190 minutes, is 0.00000900
Probability of taking 6 tries, 228 minutes, is 0.00000050
Average time to get NEW item #2 = 40 minutes (0.7 hours) BEYOND what it takes to get NEW item #1
Average time to get 2 UNIQUE items = 78 minutes (1 hours)
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TIME TO GET UNIQUE ITEM #3 (HAVE 2 UNIQUE ITEMS, MISSING 16 UNIQUE ITEMS)
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At this point, you have 2 items, and you're looking for your third.
* The probability of it taking ONE try to get a new unlock is 16/18.
* The probability of it taking TWO tries to get a new unlock is 16/18 * 2/18 (the probability it's good, times the probability that the first was bad)
* The probability of it taking THREE tries to get a new unlock is 16/18 * 2/18 * 2/18 (the probability it's good, times the probability that the first and second were both bad)
You see where this is going. Here's the the numbers for this numbers:
Probability of taking 1 tries, 38 minutes, is 0.88888889
Probability of taking 2 tries, 76 minutes, is 0.09876543
Probability of taking 3 tries, 114 minutes, is 0.01097394
Probability of taking 4 tries, 152 minutes, is 0.00121933
Probability of taking 5 tries, 190 minutes, is 0.00013548
Probability of taking 6 tries, 228 minutes, is 0.00001505
Probability of taking 7 tries, 266 minutes, is 0.00000167
Probability of taking 8 tries, 304 minutes, is 0.00000019
Average time to get NEW item #3 = 43 minutes (0.7 hours) BEYOND what it takes to get NEW item #2
Average time to get 3 UNIQUE items = 121 minutes (2 hours)
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TIME TO GET UNIQUE ITEM #18 (HAVE 17 UNIQUE ITEMS, MISSING 1 UNIQUE ITEMS)
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The rest continues pretty much straighforwardly. For completeness's sake, here's the probabilities and time it takes to get the LAST of the 18 items.
Probability of taking 1 tries, 38 minutes, is 0.05555556
Probability of taking 2 tries, 76 minutes, is 0.05246914
Probability of taking 3 tries, 114 minutes, is 0.04955418
Probability of taking 4 tries, 152 minutes, is 0.04680117
Probability of taking 5 tries, 190 minutes, is 0.04420111
Probability of taking 6 tries, 228 minutes, is 0.04174549
Probability of taking 7 tries, 266 minutes, is 0.03942630
Probability of taking 8 tries, 304 minutes, is 0.03723595
Probability of taking 9 tries, 342 minutes, is 0.03516728
Probability of taking 10 tries, 380 minutes, is 0.03321355
Probability of taking 11 tries, 418 minutes, is 0.03136835
Probability of taking 12 tries, 456 minutes, is 0.02962566
Probability of taking 13 tries, 494 minutes, is 0.02797979
Probability of taking 14 tries, 532 minutes, is 0.02642536
Probability of taking 15 tries, 570 minutes, is 0.02495728
Probability of taking 16 tries, 608 minutes, is 0.02357077
Probability of taking 17 tries, 646 minutes, is 0.02226128
Probability of taking 18 tries, 684 minutes, is 0.02102454
Probability of taking 19 tries, 722 minutes, is 0.01985651
Probability of taking 20 tries, 760 minutes, is 0.01875337
Probability of taking 21 tries, 798 minutes, is 0.01771152
Probability of taking 22 tries, 836 minutes, is 0.01672755
Probability of taking 23 tries, 874 minutes, is 0.01579824
Probability of taking 24 tries, 912 minutes, is 0.01492056
Probability of taking 25 tries, 950 minutes, is 0.01409164
Probability of taking 26 tries, 988 minutes, is 0.01330877
Probability of taking 27 tries, 1026 minutes, is 0.01256939
Probability of taking 28 tries, 1064 minutes, is 0.01187109
Probability of taking 29 tries, 1102 minutes, is 0.01121159
Probability of taking 30 tries, 1140 minutes, is 0.01058872
Probability of taking 31 tries, 1178 minutes, is 0.01000046
Probability of taking 32 tries, 1216 minutes, is 0.00944488
Probability of taking 33 tries, 1254 minutes, is 0.00892016
Probability of taking 34 tries, 1292 minutes, is 0.00842460
Probability of taking 35 tries, 1330 minutes, is 0.00795657
Probability of taking 36 tries, 1368 minutes, is 0.00751453
Probability of taking 37 tries, 1406 minutes, is 0.00709706
Probability of taking 38 tries, 1444 minutes, is 0.00670278
Probability of taking 39 tries, 1482 minutes, is 0.00633040
Probability of taking 40 tries, 1520 minutes, is 0.00597871
Probability of taking 41 tries, 1558 minutes, is 0.00564656
Probability of taking 42 tries, 1596 minutes, is 0.00533286
Probability of taking 43 tries, 1634 minutes, is 0.00503659
Probability of taking 44 tries, 1672 minutes, is 0.00475678
Probability of taking 45 tries, 1710 minutes, is 0.00449252
Probability of taking 46 tries, 1748 minutes, is 0.00424293
Probability of taking 47 tries, 1786 minutes, is 0.00400721
Probability of taking 48 tries, 1824 minutes, is 0.00378459
Probability of taking 49 tries, 1862 minutes, is 0.00357434
Probability of taking 50 tries, 1900 minutes, is 0.00337576
Probability of taking 51 tries, 1938 minutes, is 0.00318822
Probability of taking 52 tries, 1976 minutes, is 0.00301110
Probability of taking 53 tries, 2014 minutes, is 0.00284381
Probability of taking 54 tries, 2052 minutes, is 0.00268582
Probability of taking 55 tries, 2090 minutes, is 0.00253661
Probability of taking 56 tries, 2128 minutes, is 0.00239569
Probability of taking 57 tries, 2166 minutes, is 0.00226259
Probability of taking 58 tries, 2204 minutes, is 0.00213690
Probability of taking 59 tries, 2242 minutes, is 0.00201818
Probability of taking 60 tries, 2280 minutes, is 0.00190606
Probability of taking 61 tries, 2318 minutes, is 0.00180017
Probability of taking 62 tries, 2356 minutes, is 0.00170016
Probability of taking 63 tries, 2394 minutes, is 0.00160570
Probability of taking 64 tries, 2432 minutes, is 0.00151650
Probability of taking 65 tries, 2470 minutes, is 0.00143225
Probability of taking 66 tries, 2508 minutes, is 0.00135268
Probability of taking 67 tries, 2546 minutes, is 0.00127753
Probability of taking 68 tries, 2584 minutes, is 0.00120656
Probability of taking 69 tries, 2622 minutes, is 0.00113952
Probability of taking 70 tries, 2660 minutes, is 0.00107622
Probability of taking 71 tries, 2698 minutes, is 0.00101643
Probability of taking 72 tries, 2736 minutes, is 0.00095996
Probability of taking 73 tries, 2774 minutes, is 0.00090663
Probability of taking 74 tries, 2812 minutes, is 0.00085626
Probability of taking 75 tries, 2850 minutes, is 0.00080869
Probability of taking 76 tries, 2888 minutes, is 0.00076376
Probability of taking 77 tries, 2926 minutes, is 0.00072133
Probability of taking 78 tries, 2964 minutes, is 0.00068126
Probability of taking 79 tries, 3002 minutes, is 0.00064341
Probability of taking 80 tries, 3040 minutes, is 0.00060767
Probability of taking 81 tries, 3078 minutes, is 0.00057391
Probability of taking 82 tries, 3116 minutes, is 0.00054202
Probability of taking 83 tries, 3154 minutes, is 0.00051191
Probability of taking 84 tries, 3192 minutes, is 0.00048347
Probability of taking 85 tries, 3230 minutes, is 0.00045661
Probability of taking 86 tries, 3268 minutes, is 0.00043124
Probability of taking 87 tries, 3306 minutes, is 0.00040729
Probability of taking 88 tries, 3344 minutes, is 0.00038466
Probability of taking 89 tries, 3382 minutes, is 0.00036329
Probability of taking 90 tries, 3420 minutes, is 0.00034311
Probability of taking 91 tries, 3458 minutes, is 0.00032404
Probability of taking 92 tries, 3496 minutes, is 0.00030604
Probability of taking 93 tries, 3534 minutes, is 0.00028904
Probability of taking 94 tries, 3572 minutes, is 0.00027298
Probability of taking 95 tries, 3610 minutes, is 0.00025782
Probability of taking 96 tries, 3648 minutes, is 0.00024349
Probability of taking 97 tries, 3686 minutes, is 0.00022997
Probability of taking 98 tries, 3724 minutes, is 0.00021719
Probability of taking 99 tries, 3762 minutes, is 0.00020512
Average time to get NEW item #18 = 668 minutes (11.1 hours) BEYOND what it takes to get NEW item #17
Average time to get 18 UNIQUE items = 2375 minutes (40 hours)
So - on AVERAGE, most users will get their 18th and final unique item after 40 hours of gameplay. This is inline with what Valve has said to us.
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Simulation data results
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But, that's an average. What this hides is the fact that there are many people who will NOT have average "luck". I simulated this to figure out these numbers.
I ran 10,000,000 simulations that use the 38 minute / 18 item mathematics to see how long it takes users to get their 18 unique items. Here are the results:
Number of simulated users who took 0-5 hours to get all 18 items = 0 (0.0000%)
Number of simulated users who took 5-15 hours to get all 18 items = 2047 (0.0205%)
Number of simulated users who took 15-25 hours to get all 18 items = 898210 (8.9821%)
Number of simulated users who took 25-35 hours to get all 18 items = 3358689 (33.5869%)
Number of simulated users who took 35-45 hours to get all 18 items = 2968883 (29.6888%)
Number of simulated users who took 45-55 hours to get all 18 items = 1514156 (15.1416%)
Number of simulated users who took 55-65 hours to get all 18 items = 739629 (7.3963%)
Number of simulated users who took 65-75 hours to get all 18 items = 308424 (3.0842%)
Number of simulated users who took 75-85 hours to get all 18 items = 125236 (1.2524%)
Number of simulated users who took 85-95 hours to get all 18 items = 48603 (0.4860%)
Number of simulated users who took 95-105 hours to get all 18 items = 21784 (0.2178%)
Number of simulated users who took 105-115 hours to get all 18 items = 8568 (0.0857%)
Number of simulated users who took 115-125 hours to get all 18 items = 3497 (0.0350%)
Number of simulated users who took 125-135 hours to get all 18 items = 1379 (0.0138%)
Number of simulated users who took 135-145 hours to get all 18 items = 505 (0.0050%)
Number of simulated users who took 145-155 hours to get all 18 items = 217 (0.0022%)
Number of simulated users who took 155-165 hours to get all 18 items = 122 (0.0012%)
Number of simulated users who took 165-175 hours to get all 18 items = 28 (0.0003%)
Number of simulated users who took 175-185 hours to get all 18 items = 18 (0.0002%)
Number of simulated users who took 185-195 hours to get all 18 items = 2 (0.0000%)
Number of simulated users who took 195-205 hours to get all 18 items = 1 (0.0000%)
Number of simulated users who took 205-215 hours to get all 18 items = 1 (0.0000%)
Number of simulated users who took 215-225 hours to get all 18 items = 1 (0.0000%)
Number of simulated users who took 225-235 hours to get all 18 items = 0 (0.0000%)
This reveals the inherent unfairness:
* A few lucky ♥♥♥♥♥♥♥s get ALL their unlocks in 5-15 hours.
* 72% of users will get ALL unlocks in 45 hours or less. 28% of users won't.
* 2% of users will take more than 85 hours to get ALL unlocks.
* About 1 in 200,000 users will take over 165 hours to get ALL unlocks! There are several MILLION copies of TF2 out there. Quite a few people will fall into this category! Better hope you're not one of them.
Anyway, that's math. I hope somebody finds this useful or entertaining.
tasty sauce
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