Estimate using Chebyshev´s inequality
Task number: 3552
Assume that we throw a fair die one hundred times. Let the random variable \(X\) equal the sum of all values thrown.
Using Chebyshev's inequality estimate the probability \(P[|X-350|\ge 50]\).
Hint
Decompose \(X\) into a sum of many independent random variables.
Chebyshev's inequality is \(P[|X-EX|\ge t] \le \frac{var(X)}{t^2}\).
Solution
If the random variable \(Y_i\) is the value of the \(i\)-th throw of the die, then \(X=Y_1+Y_2+…+Y_{100}\), and all hundred variables are mutually independent.
For each variable \(Y_i\) we have \(EY_i=\frac72\), \(var(Y_i)=\frac{35}{12}\).
\(EX=E[Y_1+Y_2+…+Y_{100}]=100\cdot \frac72=350\).
\(var(X)=\sum_{i=1}^{100} var(Y_i)=\frac{3500}{12}\).
Now by Chebyshev's inequality
\(P[|X-350|\ge 50] \le \frac{3500}{12 {\cdot} 50^2}=\frac{7}{60}\doteq 0{,}117\)
Answer
The given probability is less than 12 %.
