Introduction:
In probability theory, Chebyshev's inequality provides an upper bound on the probability that a random variable deviates from its mean by a certain amount. It is a powerful tool for understanding the spread of a random variable around its expected value. In this case, we are given that the expected value of a random variable X is 3 and the expected value of X squared is 13. We will use Chebyshev's inequality to determine the probability that X deviates from its mean by a certain amount.

Chebyshev's Inequality:
Chebyshev's inequality states that for any random variable X with finite mean μ and finite variance σ^2, the probability that X deviates from its mean by more than k standard deviations is at most 1/k^2.

Calculating Variance:
To use Chebyshev's inequality, we need to calculate the variance of X. The variance of X is defined as the expected value of the square of the deviation of X from its mean:
Var(X) = E((X - E(X))^2)

Given that E(X) = 3 and E(X^2) = 13, we can calculate the variance as follows:
Var(X) = E((X - 3)^2) = E(X^2 - 6X + 9) = E(X^2) - 6E(X) + 9 = 13 - 6(3) + 9 = 4

Applying Chebyshev's Inequality:
Now that we have the variance of X, we can apply Chebyshev's inequality to determine the probability that X deviates from its mean by a certain amount.

Let's say we want to find the probability that X deviates from its mean by more than k standard deviations. Using Chebyshev's inequality, we have:
P(|X - E(X)| > kσ) ≤ 1/k^2

In our case, since the standard deviation is the square root of the variance, σ = sqrt(Var(X)) = sqrt(4) = 2. Therefore, we can rewrite the inequality as:
P(|X - 3| > 2k) ≤ 1/k^2

Determining the Probability:
Let's say we want to find the probability that X deviates from its mean by more than 3 standard deviations. In this case, k = 3, and the inequality becomes:
P(|X - 3| > 6) ≤ 1/3^2 = 1/9

Since the probability of any event is always between 0 and 1, we know that P(|X - 3| > 6) ≤ 1/9 is a valid inequality. Therefore, we can conclude that the probability that X deviates from its mean by more than 3 standard deviations is at most 1/9.

Conclusion:
Using Chebyshev's inequality, we determined that the probability that the random variable X deviates from its mean by more than 3 standard deviations is at most 1/9. This inequality provides a general upper bound on the probability of deviations and allows us to quantify the spread of a random variable around its expected value. By calculating the variance of X and applying Chebyshev's