Math Problem Statement
Solution
This problem requires the use of Chebyshev's inequality to find an upper bound on the probability that the number of heads deviates from the expected value by 2 or more.
Let’s break this down:
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The number of heads follows a binomial distribution: where is the number of trials (coin flips), and is the probability of getting a head in each flip.
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The expected value of is:
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The variance of is: So, the standard deviation is:
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We need to find the probability that deviates from its mean by 2 or more:
Using Chebyshev’s inequality: Here, the deviation corresponds to .
Substituting :
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Finally, applying Chebyshev’s inequality:
Therefore, the upper bound on the probability is approximately 0.63.
Would you like more details on any part of this solution? Here are five related questions to deepen your understanding:
- How would the upper bound change if the deviation was 3 instead of 2?
- What does Chebyshev's inequality tell us about distributions beyond the binomial?
- How could you calculate the exact probability using the binomial distribution?
- Can Chebyshev's inequality be applied to any distribution? Why or why not?
- What is the significance of the standard deviation in Chebyshev’s inequality?
Tip: Chebyshev’s inequality gives a conservative estimate, meaning the actual probability might be lower than the bound. It's useful when little is known about the distribution.
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Math Problem Analysis
Mathematical Concepts
Probability
Binomial Distribution
Chebyshev's Inequality
Variance
Formulas
Expected value: E(X) = np
Variance: Var(X) = np(1 - p)
Chebyshev's inequality: P(|X - μ| ≥ kσ) ≤ 1/k²
Theorems
Chebyshev's Inequality
Suitable Grade Level
Grades 10-12
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