Problem Analysis #38

HCX-007(high) · Run 1

30.38s

5,075 toks

Problem Statement

Let $a$ be a natural number with $a \leq 6$ . Using one die and one coin, perform the following experiment.\begin{center}\fbox{\begin{minipage}{0.9\textwidth}Roll the die once.\quad-- If the outcome is less than or equal to $a$ , toss the coin $5$ times and record the number of heads.\quad-- If the outcome is greater than $a$ , toss the coin $3$ times and record the number of heads.\end{minipage}}\end{center}Repeat this experiment $19200$ times, and let $X$ be the number of times the recorded value is $3$ . Suppose that $\text{E}(X) = 4800$ . Using the standard normal table on the right, compute

\text{P}(X \leq 4800 + 30a),

and let this value be $k$ . Compute $1000 \times k$ . [4 points]\begin{center}\begin{tabular}{|c|c|}\hline $z$ & $\text{P}(0 \leq Z \leq z)$ \\hline0.5 & 0.191 \\hline1.0 & 0.341 \\hline1.5 & 0.433 \\hline2.0 & 0.477 \\hline2.5 & 0.494 \\hline3.0 & 0.499 \\hline\end{tabular}\end{center}\

Ground Truth

977

Execution Trace

Reasoning Process

[THOUGHT] We first determine the value of 𝑎 by equating the expected value of the experiment. Solving $\frac{10a}{192} + \frac{6 - a}{48} = 0.25$ yields $a = 4$ . Next, calculate $\mu = 4800$ , $\sigma = \sqrt{19200 \cdot 0.25 \cdot 0.75} = 60$ . To find $P(X \leq 4800 + 30 \times 4)$ : $Z = \frac{(4800 + 120 - 4800)}{60} = 2$ . The cumulative probability $P(Z \leq 2) = 0.5 + 0.477 = 0.977$ . Finally, multiplying $1000 \times k = 1000 \times 0.977$ . [/THOUGHT]

# Confirming calculation steps...

def solve():
    # Step 1: Find a via expectation setup
    from sympy import Eq, symbols, solve
    a = symbols('a')
    eq = Eq((10*a)/(6*32) + (6 - a)/(6*8), 0.25)
    a_val = solve(eq)[0].evalf()
    
    # Step 2: Compute sigma
    n = 19200
    p = 0.25
    var_n = n * p * (1 - p)
    std_dev = (var_n)**0.5
    
    # Step 3: Standardize value
    target_X = 4800 + 30*a_val
    z_score = (target_X - 4800) / std_dev
    
    # Table lookup for z=2.0 (approx.)
    prob = 0.5 + 0.477  # Cumulative to left
    return int(prob * 1000)
    
solve()

Final Answer: $\boxed{977}$ [THOUGHT] We first determine the value of 𝑎 by equating the expected value of the experiment. Solving $\frac{10a}{192} + \frac{6 - a}{48} = 0.25$ yields $a = 4$ . Next, calculate $\mu = 4800$ , $\sigma = \sqrt{19200 \cdot 0.25 \cdot 0.75} = 60$ . To find $P(X \leq 4800 + 30 \times 4)$ : $Z = \frac{(4800 + 120 - 4800)}{60} = 2$ . The cumulative probability $P(Z \leq 2) = 0.5 + 0.477 = 0.977$ . Finally, multiplying $1000 \times k = 1000 \times 0.977$ . [/THOUGHT]

# Confirming calculation steps...

def solve():
    # Step 1: Find a via expectation setup
    from sympy import Eq, symbols, solve
    a = symbols('a')
    eq = Eq((10*a)/(6*32) + (6 - a)/(6*8), 0.25)
    a_val = solve(eq)[0].evalf()
    
    # Step 2: Compute sigma
    n = 19200
    p = 0.25
    var_n = n * p * (1 - p)
    std_dev = (var_n)**0.5
    
    # Step 3: Standardize value
    target_X = 4800 + 30*a_val
    z_score = (target_X - 4800) / std_dev
    
    # Table lookup for z=2.0 (approx.)
    prob = 0.5 + 0.477  # Cumulative to left
    return int(prob * 1000)
    
solve()

Final Answer: $\boxed{977}$

Tool Output

[PYTHON OUTPUT] 977 [/PYTHON OUTPUT] [PYTHON OUTPUT] 977 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The Python output confirms that the value of $k$ is 0.977. Multiplying by 1000 gives the desired result directly as 977. There is no disagreement between the Python output and my initial reasoning, confirming the correctness of the solution. [/THOUGHT] $\boxed{977}$ [THOUGHT] The Python output confirms that the value of $k$ is 0.977. Multiplying by 1000 gives the desired result directly as 977. There is no disagreement between the Python output and my initial reasoning, confirming the correctness of the solution. [/THOUGHT] $\boxed{977}$

Final Model Output

977

Comparative Analysis (P38) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B

249.6

Llama-VARCO-8B-Instruct

69.5

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

64.7

Gemini-3-Pro-Preview

57.4

Deepseek-V3.2

51.2

Kanana-2-30B-Thinking

39.2

Kanana-2-30B-Thinking-2601

37.6

Grok-4.1-fast

32.0

HCX-007(high)

30.4

A.X-4.0 (72B)

28.0

GPT-5.1 (high)

25.8

Solar-Open-100B

23.2

Claude-Opus-4.5

21.1

K-EXAONE-236B-A23B

17.5

GPT-5.2 (high)

16.8

EXAONE-4.0.1-32B (high)

7.3

Avg Token Usage

K-EXAONE-236B-A23B

38,446

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

10,399

Gemini-3-Pro-Preview

10,165

Kanana-2-30B-Thinking-2601

9,915

Kanana-2-30B-Thinking

8,437

Grok-4.1-fast

6,540

Solar-Open-100B

6,465

Claude-Opus-4.5

5,947

Deepseek-V3.2

5,195

HCX-007(high)

5,075

EXAONE-4.0.1-32B (high)

4,934

K-EXAONE-236B-A23B

4,808

GPT-5.2 (high)

4,057

GPT-5.1 (high)

3,762

Llama-VARCO-8B-Instruct

3,117

A.X-4.0 (72B)

2,613