A fair coin is tossed 100 times. The expected number of heads is 50, and the standard deviation for the number of heads is \((100 \cdot 1/2 \cdot 1/2)^{1/2}=5\). What does Chebychev’s inequality tell you about the probability that the number of heads that turn up deviates from the expected number of 50 by three or more standard deviations (i.e., by at least 15)?
According to Chebychev’s inequality, \(P(|X-50|\geq15)\leq\frac{25}{15^2}\\ \frac{25}{15^2}=\frac{25}{225}=\frac{1}{9}\)
Therefore, according to Chebychev’s inequality, the probability that the number of heads is NOT between 36 and 64 (inclusive) is no greater than one-ninth.