Problem Analysis #11

38.99s

4,852 toks

Problem Statement

Three students $S_1$ , $S_2$ , and $S_3$ are given a problem to solve. Consider the following events: $A$ : At least one of $S_1$ , $S_2$ , and $S_3$ can solve the problem, $B$ : $S_1$ can solve the problem, given that neither $S_2$ nor $S_3$ can solve the problem, $C$ : $S_2$ can solve the problem and $S_3$ cannot solve the problem, $D$ : $S_3$ can solve the problem.For any event $E$ , let $P(E)$ denote the probability of $E$ . If $P(B) = \frac{1}{2}, \quad P(C) = \frac{1}{10}, \quad \text{and} \quad P(D) = \frac{1}{12},$ and $P(A) = \dfrac{p}{q}$ where $p$ and $q$ are coprime positive integers, find the value of $p^2 + q^2$ .\

Ground Truth

19441

Execution Trace

Reasoning Process

# Calculate p₁ based on given P(B) = 1/2
p_B = 1/2
p_C = 1/10
p_D = 1/12
p_3 = p_D  # S3 solves

# From event C: p_2*(1-p_3) = 1/10
p_2 = p_C / (1 - p_3)
print("p2:", p_2)

# From event B: p1*(1-p_2)*(1-p3) = 1/2
p1_numerator = p_B
denominator = (1 - p_2) * (1 - p_3)
p_1 = p1_numerator / denominator
print("p1:", p_1)

# Probability that all fail: (1-p1)*(1-p2)*(1-p3)
all_fail_prob = (1 - p_1) * (1 - p_2) * (1 - p_3)
P_A = 1 - all_fail_prob

# Final fractional calculation
numerator, denominator = P_A.as_integer_ratio()
print(P_A, '=', f'{numerator}/{denominator}')
print(numerator**2 + denominator**2)

Tool Output

[PYTHON OUTPUT] p2: 0.1090909090909091 p1: 0.6122448979591837 0.6833333333333333 = 3077459745369839/4503599627370496 29753168088023464718346142171937 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] Based on the Python output, the computations correctly derived the numerator and denominator for P(A) as ( \frac{3077459745369839}{4503599627370496} ). Squaring these numbers and adding yields the requested sum. Final Answer: \boxed{29753168088023464718346142171937} \end{think}][THOUGHT] Based on the Python output, the computations correctly derived the numerator and denominator for P(A) as ( \frac{3077459745369839}{4503599627370496} ). Squaring these numbers and adding yields the requested sum. Final Answer: \boxed{29753168088023464718346142171937} \end{think}]

Final Model Output

29753168088023464718346142171937

Comparative Analysis (P11) (Average across runs)

Avg Time Taken

Gemini-3-Pro-Preview

228.5

Llama-VARCO-8B-Instruct

51.6

Deepseek-V3.2

43.4

HCX-007(high)

39.0

Solar-Pro-2 (31B)(high)

33.7

A.X-4.0 (72B)

26.7

GPT-5.1 (high)

18.1

Grok-4.1-fast

18.1

Claude-Opus-4.5

16.1

EXAONE-4.0.1-32B (high)

5.8

Avg Token Usage

Gemini-3-Pro-Preview

10703.0

Grok-4.1-fast

8474.0

Solar-Pro-2 (31B)(high)

6509.0

A.X-4.0 (72B)

5008.0

Deepseek-V3.2

4986.0

HCX-007(high)

4852.0

EXAONE-4.0.1-32B (high)

4557.0

Claude-Opus-4.5

4549.0

GPT-5.1 (high)

3536.0

Llama-VARCO-8B-Instruct

2394.0