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 total prize (X) consists of two independent contributions:
- 100 units if the first draw is crimson,
- 200 units if the fourth draw is crimson.
Since the draws are without replacement, each position in the sequence of draws has the same probability of being a crimson ball, namely (\frac{4}{9}).
Thus
[
E[X]=100\cdot\frac{4}{9}+200\cdot\frac{4}{9}= \frac{400}{3}.
]
Writing (E[X]=\frac{p}{q}) with (\gcd(p,q)=1) gives (p=400,;q=3).
We need (p\cdot q^{2}=400\cdot 3^{2}=400\cdot9=3600).
[/THOUGHT]
[PYTHON] from sympy import Rational
probability that a given draw is crimson
p_crimson = Rational(4, 9)
expected prize
E = 100 * p_crimson + 200 * p_crimson print(E) # expected value as a rational number
extract numerator and denominator
p = E.numerator q = E.denominator result = p * q**2 print(result) # p * q^2 [/PYTHON]
[THOUGHT] The Python output confirms the expected value is (\frac{400}{3}), so (p = 400) and (q = 3). Calculating (p \cdot q^2 = 400 \cdot 3^2 = 3600) gives the final result. [/THOUGHT]
\boxed{3600}