Suppose the following conditions hold. A prize-drawing event uses a box containing 4 crimson balls and 5 blue balls. You are given four opportunities to draw a ball, without replacement.Under these conditions, you receive a prize of 100(10,000-won units) if you draw a crimson ball on the first draw, and 200(10,000-won units) if you draw a crimson ball on the fourth draw. Let be the random variable representing the total prize money obtained after drawing four balls from the box. If the expected value of is written aswhere and are relatively prime positive integers, find the value ofoindent
[THOUGHT] The expected value of (X) is (E[X] = 100 \cdot P(\text{first red}) + 200 \cdot P(\text{fourth red})). Since the draws are without replacement, the probability of drawing a red ball on any specific draw is the same due to symmetry. Thus, (P(\text{first red}) = P(\text{fourth red}) = \frac{4}{9}). Substituting, (E[X] = 100 \cdot \frac{4}{9} + 200 \cdot \frac{4}{9} = \frac{1200}{9} = \frac{400}{3}). Here, (p = 400) and (q = 3), so (p \cdot q^2 = 400 \cdot 9 = 3600). [/THOUGHT] [PYTHON] from fractions import Fraction num = 1200 den = 9 frac = Fraction(num, den) p = frac.numerator q = frac.denominator result = p * q * q print(result) [/PYTHON] [THOUGHT] \boxed{3600}
[THOUGHT] The Python output confirms that p * q^2 = 3600, which matches the expected value calculation. The fraction 400/3 reduces correctly to coprime integers, and the product p * q^2 is 3600. [/THOUGHT] \boxed{3600}