## Unveiling the Formula

Embark on a journey through Chebyshev’s theorem with our user-friendly calculator. Here, we decipher the mathematical marvel behind two similar yet distinct formulas. Dive into the notation:

**P**: Probability of an event (specified within brackets).**X**: Random variable describing our experiment (e.g., the count of sixes in dice rolls).**E(X)**: Expected value of our event.**σ²**: Variance, showcasing the event’s potential divergence.**k**: Boundary of the result, indicating how far we expect to deviate.

**Formula Insights:**

- Probability of X deviating from E(X) by at least k is at most σ² / k².
- Probability of X deviating from E(X) by at least k × σ is at most 1 / k².

## Unraveling Chebyshev’s Rule

Pafnuty Chebyshev, a Russian mathematician, guides us through the realm of probability. Imagine flipping a coin a hundred times – if heads have a 0.5 probability, obtaining close to 50 heads seems likely. This informal definition aligns with the expected value (E(X)), indicating typical outcomes.

**Probability Precision:** Chebyshev’s theorem asserts that extreme results are unlikely. For instance, obtaining 25 heads in 100 coin tosses has a probability of at most 4%, and for 10 heads, it drops to 1.56%.

## Navigating Chebyshev’s Inequality

### 1. The Magic Trick Example

Picture yourself as “Chebyshev the Magnificent,” impressing with a card magic trick. Pulling 20 cards, claiming at least ten clubs, involves calculated risk:

**X**: Number of pulled club cards.**E(X)**: Expected clubs (20 × 1/4 = 5).**k**: Deviation from expected (10 – 5 = 5).**σ²**: Variance = 20 × 1/4 × 3/4 = 3.75.

According to Chebyshev’s formula, the probability is at most 15%, making the trick a risky bet.

### 2. Betting on Basketball Scores

Imagine betting on a basketball team scoring between 60 and 80 points, with an expected value around 70. Adjusting Chebyshev’s formula for proximity:

**k**: Deviation limit (e.g., 11 points).**σ²**: Variance (e.g., 20).

The probability of staying within the desired range is at least 83%, suggesting a favorable bet.

## Pros and Cons

Chebyshev’s rule aids in assessing the likelihood of events but has limitations. If the right-side fraction exceeds one, it simply indicates a maximum probability of 100%, offering a binary outcome.

In summary, Chebyshev’s theorem guides us through the maze of probability, unveiling the magic behind expected values and calculated risks. So, whether you’re a mathematician or a magic enthusiast, Chebyshev has something intriguing for you!

A: The probability in Bertrand’s paradox is not precisely defined without specifying the chord sampling method. The original question, regarding the probability of a random chord being longer than the triangle’s side, requires additional details for a definitive answer.