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an urn contains six ball of each of the three colors: red, blue, and green

an urn contains six ball of each of the three colors: red, blue, and green.

An urn contains six balls of each of the three colors: red, blue, green. Find the expected number of different colors obtained when three balls are drawn:

a. with replacement; b. without replacement

a) G.F.
When replacing, the G.F. is $$(x+y+z)^3$$

The types equation is $$ 6 + 18 + 3 = 27$$

The average # of colors is $$ {3.6 + 2.18 + 1.3 \over 27 } = {19 \over 9} $$

b) e.g.f. by counters
Species formula is:

$$\mathit 3 (rX + gY+ bZ) \cdot \mathit {15} (X+Y+Z) $$

$$coeff ({x^6\over 6!}{y^6\over6!}{z^6\over6!}) = 20(b^3+g^3+r^3)+90b^2g+90b^2r+90bg^2+90br^2+90g^2r+90gr^2 +216bgr$$

the type equation is $$ 60 + 540 + 216 = 816$$

The average # of colors is $${ 3.216 + 2.540 + 1.60 \over 816 } \approx 2.1912 $$

b) e.g.f. by combs
Let us take a card deck that contains only three colors : spade, diamond and heart, and suits from 1 to 6.

1) Define a poker hand as a hand of three cards. There are several types of hands, full color, pair and tricolor. Let count the hands without replacement :

tricolor : $$ {6 \choose 1} {6 \choose 1} {6 \choose 1} = 216 $$

pair : $$ {3 \choose 1} {2 \choose 1} {6 \choose 1} {6 \choose 2} = 540 $$

full color : $$ {3 \choose 1} {6 \choose 3}= 60 $$

their sum is $$ {18 \choose 3} = 816 $$, the average # of colors is $${ 216.3 + 540.3 + 60.1 \over 816 }\approx 2.1912  $$