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 as
where and are relatively prime positive integers, find the value of
oindent
[THOUGHT] I will compute the expected prize using linearity of expectation and verify with Python by enumerating all permutations of draws from the 9 balls (4 crimson, 5 blue). Then I will express the expected value as p/q in lowest terms and compute p*q^2. [/THOUGHT] [PYTHON] import itertools from fractions import Fraction
Represent crimson as 'C' and blue as 'B'
balls = ['C'] * 4 + ['B'] * 5
total_prize = Fraction(0, 1) count = 0
for draws in set(itertools.permutations(balls, 4)): # Prize: 100 if first is crimson, 200 if fourth is crimson prize = 0 if draws[0] == 'C': prize += 100 if draws[3] == 'C': prize += 200 total_prize += prize count += 1
expected_value = total_prize / count print("Expected value as Fraction:", expected_value)
Extract p and q
p = expected_value.numerator q = expected_value.denominator print("p:", p, "q:", q) print("p * q^2:", p * q * q)
[THOUGHT] Using the valid Python output, the expected value is 150 = 150/1, so p = 150 and q = 1, giving p·q² = 150.