What is the probability of drawing a sample of sizek

-with replacement

-from a set containingNdistinct values (and no other values). Each of these N values has an equal probability 1/N of being drawn

-containing only(where z <=k and z<=N)zdistinct values

Suppose for instance that the set is given by S={A,B,C} andk=2.

z=1: probability of drawing

( {A,A},{B,B} or {C,C} )

-->(1 distinct value)

prob = (3/9)

z=2: probability of drawing

({A,B} ,{B,A},{A,C},{C,A} ,{B,C} or {C,B})

-->(2 distinct values)

prob = (6/9)

Alternative example:

Suppose for instance that the set is given by S={A,B,C} andk=3.

z=1: probability of drawing

( {A,A,A},{B,B,B} or {C,C,C} )

-->(1 distinct value)

prob = (3/27)

z=2: probability of drawing

{A,B,B}, {B,A,B},{B,B,A}, {B,A,A}, {A,B,A},{A,A,B},

{A,C,C}, {C,A,C},{C,C,A}, {C,A,A}, {A,C,A},{A,A,C},

{C,B,B}, {B,C,B},{B,B,C}, {B,C,C}, {C,B,C}, or{C,C,B}

(2 distinct values)

prob = (18/27)

z=3: probability of drawing

({A,B,C}, {A,C,B}, {B,A,C} {B,C,A}, {C,A,B}, or {C,B,A})

-->(3 distinct values)

prob = (6/27)