Math Problem Statement
Solution
This problem revolves around a statistical test to assess whether a die follows the Laplace model (fair dice), based on the frequency of rolling a six over 100 trials. The main steps involve:
- Modeling the probabilities: Using the binomial distribution , as the probability of rolling a six on one throw is .
- Condition check: Thomas concludes the die is fair only if the number of sixes lies between 13 and 21, inclusive.
- Calculation of the required probability: Determine , where represents the number of sixes rolled.
Let me solve it step by step:
Step 1: Probability Distribution
The random variable , the number of sixes rolled in 100 trials, follows a binomial distribution: The probability mass function is:
Step 2: Compute
This involves summing probabilities:
For efficiency, we often use the normal approximation to the binomial distribution: where:
The range is approximated by:
Step 3: Standardization
Using the values of and :
From the standard normal table: Thus:
Final Answer
The probability that Thomas concludes the die is fair is approximately 77.18%.
Would you like to see the detailed binomial calculations or confirm this with a specific computation? Let me know! 😊
Related Questions
- What are the characteristics of the Laplace distribution for fair dice rolls?
- How does the binomial distribution apply to dice rolling experiments?
- Why is the normal approximation valid in this case?
- How would the conclusion change if the range were altered?
- What is the probability of rolling fewer than 13 sixes in this experiment?
Tip:
For binomial distributions with large and not too close to 0 or 1, the normal approximation simplifies calculations greatly!
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Binomial Distribution
Normal Approximation
Formulas
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
μ = np
σ = √(np(1-p))
P(a ≤ X ≤ b) ≈ P((a-0.5 - μ)/σ ≤ Z ≤ (b+0.5 - μ)/σ)
Theorems
Central Limit Theorem
Properties of the Normal Distribution
Suitable Grade Level
Grade 11-12 (Advanced High School) or University Level
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