What can be said about P[ 0 < X < 40 ] ?

Question

The response time of a system, X, is a random variable with a mean of 20 seconds and a
variance of 20 seconds. What can be said about P[ 0 < X < 40 ] ?

Answer 1

Use Chebyshev's inequality: If X is a random variable with expected value μ and  non-zero variance σ², then for any real number k > 0, P(|X-µ|<k)≥1-σ²/k².

With µ=20=σ²=k you get

P(|X-20|<20) ≥ 1-20/20² = .95

P(0 < X < 40) ≥ .95