[THOUGHT] We have a box with 4 crimson balls and 5 blue balls, total 9 balls. We draw 4 balls without replacement. Prize: 100 (units of 10,000 won) if crimson on first draw, 200 if crimson on fourth draw. Let X = total prize money. We need E[X] = p/q in lowest terms, then compute p * q^2.

Define indicator random variables: I1 = 1 if first draw is crimson, 0 otherwise → prize contribution 100 * I1. I4 = 1 if fourth draw is crimson, 0 otherwise → prize contribution 200 * I4. Thus X = 100 I1 + 200 I4.

Expectation: E[X] = 100 * E[I1] + 200 * E[I4].

E[I1] = P(first draw is crimson) = 4/9.

E[I4] = P(fourth draw is crimson). By symmetry, without replacement, each position equally likely to have a crimson ball. So P(crimson on fourth) = 4/9 as well. Let's verify with calculation.

Alternatively compute directly: P(4th crimson) = sum over number of crimson in first three draws? But symmetry is valid because draws are exchangeable. So E[I4] = 4/9.

Thus E[X] = 100*(4/9) + 200*(4/9) = 300*(4/9) = 1200/9 = 400/3 after simplifying? 1200/9 = 400/3 indeed (divide numerator and denominator by 3). So p=400, q=3. Then p*q^2 = 400 * 9 = 3600.

But wait: Are the draws without replacement? Yes. But is the symmetry argument valid? Yes, because the joint distribution of the sequence is exchangeable: any permutation equally likely. So probability crimson in position 4 = probability crimson in position 1 = 4/9.

Let's verify with Python simulation or exact computation to be safe.

We can compute P(4th crimson) by considering total ways to arrange 4 crimson among 9 positions? Actually, we can compute directly: total sequences of 4 draws from 9 balls without replacement: 9P4 = 9876. Number of sequences where 4th is crimson: choose which crimson ball in 4th position: 4 choices, then arrange remaining 8 balls in first three positions: 8P3 = 876. So probability = (4 * 876) / (9876) = 4/9. Good.

Thus E[X] = 100*(4/9) + 200*(4/9) = 300*(4/9) = 1200/9 = 400/3.

Check if 400 and 3 are relatively prime: gcd(400,3)=1. So p=400, q=3, pq^2 = 4009=3600.

Let's compute with Python to confirm.

[/THOUGHT] [PYTHON] import fractions

Compute expected value

E_I1 = fractions.Fraction(4, 9) E_I4 = fractions.Fraction(4, 9) E_X = 100 * E_I1 + 200 * E_I4 print("E[X] =", E_X) print("Numerator =", E_X.numerator) print("Denominator =", E_X.denominator) result = E_X.numerator * (E_X.denominator ** 2) print("p * q^2 =", result) [/PYTHON]

[THOUGHT] The Python output confirms that E[X] = 400/3, so p = 400, q = 3, and p·q² = 400·9 = 3600. [/THOUGHT] \boxed{3600}