I've drawn 11 Steam 2024 Summer Sale trading cards so far from going through the Discovery Queue. But I only got 4 unique cards; the other 7 are all duplicates.
This got me wondering? What is the chance of that happening? Should I buy a lottery ticket with my luck?
Being thoroughly nerd sniped, I've worked out the math (see the end of this post) and verified it via simulation.
Here's a probability table for the number of cards drawn vs. the number of unique cards got towards a complete badge set:
| Drawn | Got 1 | Got 2 | Got 3 | Got 4 | Got 5 | Got 6 | Got 7 | Got 8 | Got 9 | Got 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 100.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% |
| 2 | 10.000% | 90.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% |
| 3 | 1.000% | 27.000% | 72.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% |
| 4 | 0.100% | 6.300% | 43.200% | 50.400% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% |
| 5 | 0.010% | 1.350% | 18.000% | 50.400% | 30.240% | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% |
| 6 | 0.001% | 0.279% | 6.480% | 32.760% | 45.360% | 15.120% | 0.000% | 0.000% | 0.000% | 0.000% |
| 7 | 0.000% | 0.057% | 2.167% | 17.640% | 42.336% | 31.752% | 6.048% | 0.000% | 0.000% | 0.000% |
| 8 | 0.000% | 0.011% | 0.696% | 8.573% | 31.752% | 40.219% | 16.934% | 1.814% | 0.000% | 0.000% |
| 9 | 0.000% | 0.002% | 0.218% | 3.916% | 21.020% | 40.008% | 27.942% | 6.532% | 0.363% | 0.000% |
| 10 | 0.000% | 0.000% | 0.067% | 1.719% | 12.860% | 34.514% | 35.562% | 13.608% | 1.633% | 0.036% |
| 11 | 0.000% | 0.000% | 0.021% | 0.735% | 7.461% | 27.138% | 38.699% | 21.555% | 4.191% | 0.200% |
| 12 | 0.000% | 0.000% | 0.006% | 0.308% | 4.171% | 20.014% | 37.945% | 28.854% | 8.083% | 0.619% |
| 13 | 0.000% | 0.000% | 0.002% | 0.128% | 2.271% | 14.094% | 34.567% | 34.467% | 13.046% | 1.427% |
| 14 | 0.000% | 0.000% | 0.001% | 0.052% | 1.212% | 9.592% | 29.834% | 37.943% | 18.634% | 2.732% |
| 15 | 0.000% | 0.000% | 0.000% | 0.021% | 0.637% | 6.361% | 24.721% | 39.305% | 24.360% | 4.595% |
| 16 | 0.000% | 0.000% | 0.000% | 0.009% | 0.331% | 4.135% | 19.849% | 38.860% | 29.785% | 7.031% |
| 17 | 0.000% | 0.000% | 0.000% | 0.003% | 0.171% | 2.647% | 15.548% | 37.043% | 34.578% | 10.009% |
| 18 | 0.000% | 0.000% | 0.000% | 0.001% | 0.088% | 1.674% | 11.943% | 34.299% | 38.529% | 13.467% |
| 19 | 0.000% | 0.000% | 0.000% | 0.001% | 0.045% | 1.048% | 9.029% | 31.022% | 41.536% | 17.320% |
| 20 | 0.000% | 0.000% | 0.000% | 0.000% | 0.023% | 0.651% | 6.740% | 27.526% | 43.587% | 21.474% |
| 21 | 0.000% | 0.000% | 0.000% | 0.000% | 0.011% | 0.402% | 4.978% | 24.043% | 44.733% | 25.832% |
| 22 | 0.000% | 0.000% | 0.000% | 0.000% | 0.006% | 0.247% | 3.646% | 20.728% | 45.068% | 30.306% |
| 23 | 0.000% | 0.000% | 0.000% | 0.000% | 0.003% | 0.151% | 2.651% | 17.676% | 44.707% | 34.813% |
| 24 | 0.000% | 0.000% | 0.000% | 0.000% | 0.001% | 0.092% | 1.916% | 14.936% | 43.772% | 39.283% |
| 25 | 0.000% | 0.000% | 0.000% | 0.000% | 0.001% | 0.056% | 1.378% | 12.523% | 42.382% | 43.660% |
| 26 | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.034% | 0.987% | 10.432% | 40.648% | 47.899% |
| 27 | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.021% | 0.704% | 8.642% | 38.670% | 51.963% |
| 28 | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.012% | 0.501% | 7.125% | 36.531% | 55.830% |
| 29 | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.008% | 0.356% | 5.850% | 34.303% | 59.483% |
| 30 | 0.000% | 0.000% | 0.000% | 0.000% | 0.000% | 0.005% | 0.252% | 4.787% | 32.043% | 62.914% |
Math
P = S(n, k) C(m, k) k! / mn where:
- k = number of unique cards received (without duplicates),
- n = number of cards drawn (including all duplicates), and
- m = number of cards in the badge set.
- S() are the Stirling numbers of the second kind, and
- C() are the usual binomial coefficients.
Posted by Boojum