What is the probability of being selected for execution?

Question

Thirty Arkton hostages in a Brumton prison in occupied Arkland need to select three of them to be executed by their captives in retribution to the killing of three Brums by the Ark army. They tear down an old letter into thirty pieces, draw a cross on three of them, and alphabetically one after the other draw them out of a shoe. I need the answer of the following,

1. What is the probability you are chosen to be executed if you draw 1st from the shoe? Second? Third?

Answer 1

Parts (2) and (3) are not math questions.

For (1), you're right, you have a $\frac3{30}$ chance of drawing a marked piece regardless of your position in line.

Here's an attempt to elaborate on the "obvious" symmetry argument: The shoe ensures that all $30!$ permutations of the 30 pieces are equally likely, so by symmetry, each of the $\binom{30}3$ sets of 3 positions is equally likely to be the set that gets the marked pieces. For a given position $i$, there are $\binom{29}2$ such sets that include $i$, so the probability that $i$ ends up in the marked set is $\binom{29}2/\binom{30}3=\frac3{30}$.

Alternatively, you could compute it using conditional probabilities: Let $E_i$ be the event that the $i$th piece drawn is marked. Clearly $P(E_1)=\frac3{30}$. Then $$P(E_2)=P(E_2|E_1)P(E_1)+P(E_2|\lnot E_1)P(\lnot E_1)=\frac2{29}\frac3{30}+\frac3{29}\frac{27}{30}=\frac3{30}.$$ And so on.

Answer 2

Consider the thirty pieces laid out in a line in order of being drawn. The probability of the $i$th piece having a cross is $\frac{3}{30}.$

Nothing else needs to be said.