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THE FEUTEX TOOLKIT

Coupon Collection Expectation

Calculate the expected draw count and variance needed to collect every uniformly distributed coupon type.

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02 Your result
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FormulaExpected Draws = n*((n>=1?1/1:0)+(n>=2?1/2:0)+(n>=3?1/3:0)+(n>=4?1/4:0)+(n>=5?1/5:0)+(n>=6?1/6:0)+(n>=7?1/7:0)+(n>=8?1/8:0)+(n>=9?1/9:0)+(n>=10?1/10:0)+(n>=11?1/11:0)+(n>=12?1/12:0)+(n>=13?1/13:0)+(n>=14?1/14:0)+(n>=15?1/15:0)+(n>=16?1/16:0)+(n>=17?1/17:0)+(n>=18?1/18:0)+(n>=19?1/19:0)+(n>=20?1/20:0)+(n>=21?1/21:0)+(n>=22?1/22:0)+(n>=23?1/23:0)+(n>=24?1/24:0)+(n>=25?1/25:0)+(n>=26?1/26:0)+(n>=27?1/27:0)+(n>=28?1/28:0)+(n>=29?1/29:0)+(n>=30?1/30:0)+(n>=31?1/31:0)+(n>=32?1/32:0)+(n>=33?1/33:0)+(n>=34?1/34:0)+(n>=35?1/35:0)+(n>=36?1/36:0)+(n>=37?1/37:0)+(n>=38?1/38:0)+(n>=39?1/39:0)+(n>=40?1/40:0)+(n>=41?1/41:0)+(n>=42?1/42:0)+(n>=43?1/43:0)+(n>=44?1/44:0)+(n>=45?1/45:0)+(n>=46?1/46:0)+(n>=47?1/47:0)+(n>=48?1/48:0)+(n>=49?1/49:0)+(n>=50?1/50:0)+(n>=51?1/51:0)+(n>=52?1/52:0)+(n>=53?1/53:0)+(n>=54?1/54:0)+(n>=55?1/55:0)+(n>=56?1/56:0)+(n>=57?1/57:0)+(n>=58?1/58:0)+(n>=59?1/59:0)+(n>=60?1/60:0)+(n>=61?1/61:0)+(n>=62?1/62:0)+(n>=63?1/63:0)+(n>=64?1/64:0)+(n>=65?1/65:0)+(n>=66?1/66:0)+(n>=67?1/67:0)+(n>=68?1/68:0)+(n>=69?1/69:0)+(n>=70?1/70:0)+(n>=71?1/71:0)+(n>=72?1/72:0)+(n>=73?1/73:0)+(n>=74?1/74:0)+(n>=75?1/75:0)+(n>=76?1/76:0)+(n>=77?1/77:0)+(n>=78?1/78:0)+(n>=79?1/79:0)+(n>=80?1/80:0)+(n>=81?1/81:0)+(n>=82?1/82:0)+(n>=83?1/83:0)+(n>=84?1/84:0)+(n>=85?1/85:0)+(n>=86?1/86:0)+(n>=87?1/87:0)+(n>=88?1/88:0)+(n>=89?1/89:0)+(n>=90?1/90:0)+(n>=91?1/91:0)+(n>=92?1/92:0)+(n>=93?1/93:0)+(n>=94?1/94:0)+(n>=95?1/95:0)+(n>=96?1/96:0)+(n>=97?1/97:0)+(n>=98?1/98:0)+(n>=99?1/99:0)+(n>=100?1/100:0)); Variance = n**2*((n>=1?1/1:0)+(n>=2?1/4:0)+(n>=3?1/9:0)+(n>=4?1/16:0)+(n>=5?1/25:0)+(n>=6?1/36:0)+(n>=7?1/49:0)+(n>=8?1/64:0)+(n>=9?1/81:0)+(n>=10?1/100:0)+(n>=11?1/121:0)+(n>=12?1/144:0)+(n>=13?1/169:0)+(n>=14?1/196:0)+(n>=15?1/225:0)+(n>=16?1/256:0)+(n>=17?1/289:0)+(n>=18?1/324:0)+(n>=19?1/361:0)+(n>=20?1/400:0)+(n>=21?1/441:0)+(n>=22?1/484:0)+(n>=23?1/529:0)+(n>=24?1/576:0)+(n>=25?1/625:0)+(n>=26?1/676:0)+(n>=27?1/729:0)+(n>=28?1/784:0)+(n>=29?1/841:0)+(n>=30?1/900:0)+(n>=31?1/961:0)+(n>=32?1/1024:0)+(n>=33?1/1089:0)+(n>=34?1/1156:0)+(n>=35?1/1225:0)+(n>=36?1/1296:0)+(n>=37?1/1369:0)+(n>=38?1/1444:0)+(n>=39?1/1521:0)+(n>=40?1/1600:0)+(n>=41?1/1681:0)+(n>=42?1/1764:0)+(n>=43?1/1849:0)+(n>=44?1/1936:0)+(n>=45?1/2025:0)+(n>=46?1/2116:0)+(n>=47?1/2209:0)+(n>=48?1/2304:0)+(n>=49?1/2401:0)+(n>=50?1/2500:0)+(n>=51?1/2601:0)+(n>=52?1/2704:0)+(n>=53?1/2809:0)+(n>=54?1/2916:0)+(n>=55?1/3025:0)+(n>=56?1/3136:0)+(n>=57?1/3249:0)+(n>=58?1/3364:0)+(n>=59?1/3481:0)+(n>=60?1/3600:0)+(n>=61?1/3721:0)+(n>=62?1/3844:0)+(n>=63?1/3969:0)+(n>=64?1/4096:0)+(n>=65?1/4225:0)+(n>=66?1/4356:0)+(n>=67?1/4489:0)+(n>=68?1/4624:0)+(n>=69?1/4761:0)+(n>=70?1/4900:0)+(n>=71?1/5041:0)+(n>=72?1/5184:0)+(n>=73?1/5329:0)+(n>=74?1/5476:0)+(n>=75?1/5625:0)+(n>=76?1/5776:0)+(n>=77?1/5929:0)+(n>=78?1/6084:0)+(n>=79?1/6241:0)+(n>=80?1/6400:0)+(n>=81?1/6561:0)+(n>=82?1/6724:0)+(n>=83?1/6889:0)+(n>=84?1/7056:0)+(n>=85?1/7225:0)+(n>=86?1/7396:0)+(n>=87?1/7569:0)+(n>=88?1/7744:0)+(n>=89?1/7921:0)+(n>=90?1/8100:0)+(n>=91?1/8281:0)+(n>=92?1/8464:0)+(n>=93?1/8649:0)+(n>=94?1/8836:0)+(n>=95?1/9025:0)+(n>=96?1/9216:0)+(n>=97?1/9409:0)+(n>=98?1/9604:0)+(n>=99?1/9801:0)+(n>=100?1/10000:0))-n*((n>=1?1/1:0)+(n>=2?1/2:0)+(n>=3?1/3:0)+(n>=4?1/4:0)+(n>=5?1/5:0)+(n>=6?1/6:0)+(n>=7?1/7:0)+(n>=8?1/8:0)+(n>=9?1/9:0)+(n>=10?1/10:0)+(n>=11?1/11:0)+(n>=12?1/12:0)+(n>=13?1/13:0)+(n>=14?1/14:0)+(n>=15?1/15:0)+(n>=16?1/16:0)+(n>=17?1/17:0)+(n>=18?1/18:0)+(n>=19?1/19:0)+(n>=20?1/20:0)+(n>=21?1/21:0)+(n>=22?1/22:0)+(n>=23?1/23:0)+(n>=24?1/24:0)+(n>=25?1/25:0)+(n>=26?1/26:0)+(n>=27?1/27:0)+(n>=28?1/28:0)+(n>=29?1/29:0)+(n>=30?1/30:0)+(n>=31?1/31:0)+(n>=32?1/32:0)+(n>=33?1/33:0)+(n>=34?1/34:0)+(n>=35?1/35:0)+(n>=36?1/36:0)+(n>=37?1/37:0)+(n>=38?1/38:0)+(n>=39?1/39:0)+(n>=40?1/40:0)+(n>=41?1/41:0)+(n>=42?1/42:0)+(n>=43?1/43:0)+(n>=44?1/44:0)+(n>=45?1/45:0)+(n>=46?1/46:0)+(n>=47?1/47:0)+(n>=48?1/48:0)+(n>=49?1/49:0)+(n>=50?1/50:0)+(n>=51?1/51:0)+(n>=52?1/52:0)+(n>=53?1/53:0)+(n>=54?1/54:0)+(n>=55?1/55:0)+(n>=56?1/56:0)+(n>=57?1/57:0)+(n>=58?1/58:0)+(n>=59?1/59:0)+(n>=60?1/60:0)+(n>=61?1/61:0)+(n>=62?1/62:0)+(n>=63?1/63:0)+(n>=64?1/64:0)+(n>=65?1/65:0)+(n>=66?1/66:0)+(n>=67?1/67:0)+(n>=68?1/68:0)+(n>=69?1/69:0)+(n>=70?1/70:0)+(n>=71?1/71:0)+(n>=72?1/72:0)+(n>=73?1/73:0)+(n>=74?1/74:0)+(n>=75?1/75:0)+(n>=76?1/76:0)+(n>=77?1/77:0)+(n>=78?1/78:0)+(n>=79?1/79:0)+(n>=80?1/80:0)+(n>=81?1/81:0)+(n>=82?1/82:0)+(n>=83?1/83:0)+(n>=84?1/84:0)+(n>=85?1/85:0)+(n>=86?1/86:0)+(n>=87?1/87:0)+(n>=88?1/88:0)+(n>=89?1/89:0)+(n>=90?1/90:0)+(n>=91?1/91:0)+(n>=92?1/92:0)+(n>=93?1/93:0)+(n>=94?1/94:0)+(n>=95?1/95:0)+(n>=96?1/96:0)+(n>=97?1/97:0)+(n>=98?1/98:0)+(n>=99?1/99:0)+(n>=100?1/100:0))
Sources and model references

Worked example

Inputs

Equally likely coupon types
10 categories

Expected results

Expected Draws
29.28968254 draws
Variance
125.6870906 draws²

Load the example in the workspace to reproduce these results. Display rounding does not change the underlying calculation.

How to use this tool

  1. Enter the labeled inputs using consistent units.
  2. Check the stated assumptions and calculate.
  3. Review each output and compare another input scenario.

Try this example

{
  "n": 10
}

Accuracy and scope

Independent uniform coupon draws with replacement; exact finite sums, at most 100 types. Finite floating-point arithmetic; results can lose precision near singular or extreme inputs.

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