[THOUGHT] Let me analyze this problem step by step.

We have a box with 4 crimson balls and 5 blue balls (9 total). We draw 4 balls without replacement.

Prize structure:

• 100 (in 10,000-won units) if we draw a crimson ball on the first draw
• 200 (in 10,000-won units) if we draw a crimson ball on the fourth draw

Let X be the total prize money.

X can be:

• 0 (if neither first nor fourth draw is crimson)
• 100 (if only first draw is crimson)
• 200 (if only fourth draw is crimson)
• 300 (if both first and fourth draws are crimson)

I need to calculate the probability of each case and find E[X].

Let me use indicator random variables:

• Let I₁ = 1 if first draw is crimson, 0 otherwise
• Let I₄ = 1 if fourth draw is crimson, 0 otherwise

Then X = 100·I₁ + 200·I₄

E[X] = 100·E[I₁] + 200·E[I₄] = 100·P(first is crimson) + 200·P(fourth is crimson)

P(first is crimson) = 4/9

P(fourth is crimson) = ?

By symmetry in drawing without replacement, the probability that any specific position has a crimson ball is the same as the proportion of crimson balls in the box.

So P(fourth is crimson) = 4/9

Therefore E[X] = 100·(4/9) + 200·(4/9) = 400/9 + 800/9 = 1200/9 = 400/3

Let me verify this with Python by computing all possibilities. [/THOUGHT] [PYTHON] from itertools import permutations from fractions import Fraction

4 crimson balls (C) and 5 blue balls (B)

Total 9 balls, we draw 4 without replacement

balls = ['C'] * 4 + ['B'] * 5

Count all possible 4-ball draws (ordered)

total_outcomes = 0 total_prize = 0

We need to consider all permutations of choosing 4 balls from 9

from itertools import permutations

Generate all possible sequences of 4 draws

all_draws = list(permutations(balls, 4))

for draw in all_draws: prize = 0 if draw[0] == 'C': # First draw is crimson prize += 100 if draw[3] == 'C': # Fourth draw is crimson prize += 200 total_prize += prize total_outcomes += 1

expected_value = Fraction(total_prize, total_outcomes) print(f"Total outcomes: {total_outcomes}") print(f"Total prize sum: {total_prize}") print(f"Expected value: {expected_value}") print(f"p = {expected_value.numerator}, q = {expected_value.denominator}") print(f"p * q^2 = {expected_value.numerator * expected_value.denominator**2}") [/PYTHON]

[THOUGHT] The Python calculation confirms that the expected value E[X] = 400/3, where p = 400 and q = 3 are relatively prime positive integers. Therefore, p · q² = 400 × 3² = 400 × 9 = 3600. [/THOUGHT]

\boxed{3600}