Calculate the probability that when n balls are in the urn, i of them are white

Question

The Polya urn model for contagion is as follows. We start with an urn which contains one white ball and one black ball. At each second we choose a ball at random from the urn and replace it together with one more ball of the same colour. Calculate the probability that when n balls are in the urn, i of them are white.

Answer 1

0 sec 0 BW (1/2) (one white ball out of 2 balls)

1 sec 0.1 B: BW→BBW (1/3)

1 sec0.2 W: BW→BWW (2/3)

2 sec 0.1.1 B: BBW→BBBW (1/4)

2 sec 0.1.2 W: BBW→BBWW (2/4)

2 sec 0.2.1 B: BWW→BBWW (2/4)

2 sec 0.2.2 W: BWW→BWWW (3/4)

3 sec 0.1.1.1 B: BBBW→BBBBW (1/5) 3 sec

3 sec 0.1.1.2 W: BBBW→BBBWW (2/5)

3 sec 0.1.2.1 B: BBWW→BBBWW (2/5)

3 sec 0.1.2.2 W: BBWW→BBWWW (3/5)

3 sec 0.2.1.1 B: BBWW→BBBWW (2/5)

3 sec 0.2.1.2 W: BBWW→BBWWW (3/5)

3 sec 0.2.2.1 B: BWWW→BBWWW (3/5)

3 sec 0.2.2.2 W: BWWW→BWWWW (4/5)

The above represents a tree of consecutive outcomes in progressive seconds. The colour-coding shows how the number of outcomes doubles every second: 2, 4, 8, 16, etc.

Write the numbers from 1 to n-1 along a row.

Write Pascal's Triangle row t (t seconds) under the numbers from 1 to n-1.

Example 1: for n=5, t=n-2=3 sec:

1 2 3 4 (1 to n-1)

1 3 3 1 (Pascal row 3) 8 outcomes.

Example 2: for n=6, t=n-2=4 sec:

1 2 3 4 5 (1 to n-1)

1 4 6 4 1 (Pascal row 4) 16 outcomes.

The Pascal numbers are ^tC_r (also represented by C(t,r)), being the coefficients in the binomial expansion of (a+b)^t). Number of outcomes is 2^t (t can be replaced by n-2).

Probability of n-1 white balls is 1/2^n-2, probability of 1 white ball is also 1/2^n-2.

Probability of n-2 or 2 white balls is (n-2)/2^n-2; probability of n-3 or 3 white balls is (n-2)(n-3)/2! = ^tC₂.

Probability of i white balls is ^n-2C_i-1/2^n-2, where 1≤i≤n-1, for integer i. Example: n=6, 2^n-2=16:

i     p     i-1  ^n-2C_i-1

1  1/16   0   ⁴C₀=1

2  1/4     1   ⁴C₁=(n-2)/1!=4

3  3/8     2   ⁴C₂=(n-2)(n-3)/2!=4×3/2=6

4  1/4     3   ⁴C₃=(n-2)(n-3)(n-4)/3!=4×3×2/6=4

5  1/16   4   ⁴C₄=(n-2)(n-3)(n-4)(n-5)/4!=4×3×2×1/24=1