The problem is from DeGroot 3rd edition, section 1.7,

problem 10: n=100 balls, r red balls. The balls

are chosen one at a time without replacement.

a) what is the pr that a red ball is chosen on the

first selection? b) what is the pr a red ball is

chosen on the 50th selection? c) what is the pr

a red ball is chosen on the 100th selection?

a) Pr(red ball 1st) = r/n, easy

c) Pr(red ball 100th)? There are 100!

arrangements of the n=100 balls because sampling

is w/o replacement. Of those arrangements, the

only way to guarantee a red ball is chosen last

is to arrange the rest of the r-1 red balls in

the first 99 positions, which may be done in

P(99,r-1) ways where P(n,k) is the permutation

function. Therefore, Pr(red ball 100th) =

P(99,r-1) / 100!

b) Pr(red ball 50th)? Not so sure of this one.

I don't think I can use the above reasoning

because r might be > 50. There are a total of

100! permutations in 100 selections, 50! by the

50th selection. If r<=50 then there are P(50,r)

permutations of the red balls. To guarantee the

50th is a red ball, there are P(49,r-1) ways to

arrange the red balls. If r=50, then Pr(50th

red)=1. If r>50, then Pr(50th red)=1. For r<50

Pr(50th red) should be P(49,r-1)/50! But how to

combine the three cases for r to get one pr for

all r? I'm fairly sure I'm missing something

here.