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] The random variable X = 100 * I_first_crimson + 200 * I_fourth_crimson, where I are indicators. By linearity, E[X] = 100 * P(first crimson) + 200 * P(fourth crimson). Due to symmetry in sampling without replacement, P(first crimson) = 4/9 and P(fourth crimson) = 4/9. Thus E[X] = 300 * (4/9) = 1200/9 = 400/3. Here p=400, q=3, and p * q^2 = 400 * 9 = 3600. Verify with Python using fractions. [/THOUGHT] [PYTHON] from fractions import Fraction
p1 = Fraction(4, 9) p4 = Fraction(4, 9) ex = 100 * p1 + 200 * p4 p = ex.numerator q = ex.denominator result = p * q**2 print(result) [/PYTHON]
[THOUGHT] The Python code confirms the result is 3600. [/THOUGHT]
Explanation
The reasoning begins by defining the random variable X as the total prize money: X = 100 * I_first + 200 * I_fourth, where I_first is the indicator for drawing a crimson ball on the first draw, and I_fourth is the indicator for the fourth draw. The expected value is computed using linearity of expectation: E[X] = 100 * P(I_first = 1) + 200 * P(I_fourth = 1).
Next, the probabilities are determined. With 4 crimson balls out of 9 total balls and draws without replacement, the probability of drawing a crimson ball on the first draw is 4/9. Due to the symmetry of sampling without replacement, the probability for the fourth draw is also 4/9, as each ball is equally likely to be in any position in the sequence.
Thus, E[X] = 100 * (4/9) + 200 * (4/9) = 300 * (4/9) = 1200/9, which simplifies to 400/3. Here, 400 and 3 are relatively prime, so p = 400 and q = 3. The required value is p * q^2 = 400 * 3^2 = 400 * 9 = 3600.
To verify, Python code using the fractions module computes these values exactly: p1 = Fraction(4, 9), p4 = Fraction(4, 9), ex = 100 * p1 + 200 * p4, then extracts p and q from ex and calculates p * q**2, yielding 3600, confirming the result.